{VERSION 6 0 "IBM INTEL NT" "6.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 
1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 
0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 
0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 
0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 
102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times
" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 
261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 
0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 
2 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 
}{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 260 283 "" 1 18 0 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" 260 284 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 0 0 1 0 0 0 
0 0 0 0 }{CSTYLE "" 260 289 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 290 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
291 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 0 0 1 0 0 0 
0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 295 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
296 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 300 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
301 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 302 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 303 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 304 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 305 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
306 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 307 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 260 308 "" 1 18 0 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" 260 309 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 310 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Em
phasis" -1 311 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "N
ormal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 
1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" 
-1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 
2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 
0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple
 Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 
1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 
{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 
0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 
1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 
2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 
{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 
0 1 0 1 0 2 2 0 1 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 
0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 
260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 
-1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 46 "Simple harmonic motion and damped
 oscillations" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo,
 B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version:  27.3.2007" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 5 "load " }{TEXT 0 7 "desolve" }{TEXT -1 1 
" " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 311 7 "DE
sol.m" }{TEXT -1 32 " is required by this worksheet. " }}{PARA 0 "" 0 
"" {TEXT -1 121 "It can be read into a Maple session by a command simi
lar to the one that follows, where the file path gives its location." 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/procdrs
/DEsol.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 46 "An animation to illustrate oscillatory motion " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{GLPLOT2D 338 353 353 {PLOTDATA 2 "6%-%(ANIMATEG6$7--%
'CURVESG6%7[gm7$$!3_6L:&GM50#!#F$\"3%)***Rl#fTrI!#<7$$\"3kv(=E6g%e8!#=
$\"3ONLHy(>$oIF17$$\"3%p@X%yUt\"p#F5$\"3cdi$=!*)[kIF17$$\"3Ui>'e\\.^(R
F5$\"3mc'oPzV*fIF17$$\"37E%fm+uZ=&F5$\"3#R>'f#>=Z0$F17$$\"3!eu39e>%[qF
5$\"3<r]\"\\\\YT/$F17$$\"35;')>U&yv^)F5$\"3NI)Rjs')>.$F17$$\"3'HZi()p;
sf*F5$\"3@)H'zHQ$p,$F17$$\"3S3fgn0O)***F5$\"3))zlTLk3,IF17$$\"3)>'\\)R
U'G)o*F5$\"3A()RWeR3&)HF17$$\"3Ip\"H,'RA%o)F5$\"3KN'f9I\"zpHF17$$\"3M*
o]OFtk%zF5$\"3!GcvxupE'HF17$$\"3Uf!G<YGr1(F5$\"3;SVzx3)f&HF17$$\"3Gfp(
y6e=1'F5$\"3+^0Lriz\\HF17$$\"3?fI)zju&[\\F5$\"3V\"=')H_yT%HF17$$\"3$QG
%>z(*=PQF5$\"3J^v-\"[A&RHF17$$\"3y1Hq'e@qm#F5$\"3#>\"=%fIKa$HF17$$\"3E
6[9,j*fX\"F5$\"3%*RmP5F$>$HF17$$\"3S+D@\"ogmA#!#>$\"3q[Hnf$R!HHF17$$!3
YB9T*HMx0\"F5$\"3E$4w*H!*oEHF17$$!3vq-]'\\j2K#F5$\"3%H)z7N>*\\#HF17$$!
33]#)e%zyca$F5$\"3kp4xCQ$R#HF17$$!3w;Ko4MO7ZF5$\"34#30X&**[BHF17$$!3<U
9!p#[KPeF5$\"3&otg&p)RO#HF17$$!3Y%R#Rj*f(foF5$\"35^JpA-OCHF17$$!3#38)o
a3rhxF5$\"3&G^1Ix\"fDHF17$$!3'\\HP9rNt_)F5$\"3%z&3)4Ims#HF17$$!3%[m&*e
eegf*F5$\"3?k[k,#=;$HF17$$!3_kg:Ov$y***F5$\"3@/p=qAvOHF17$$!3!f@5YP;io
*F5$\"3g66G$H'3UHF17$$!3'z*y;Tj`i')F5$\"3sN%RO[*oYHF17$$!3D;NAg6yHsF5$
\"3?b'*3C\"\\&\\HF17$$!3#R,fo\"z7%Q&F5$\"39bt()=0$3&HF17$$!3=vpe'GNC?%
F5$\"3#RI:%[Hs]HF17$$!3#=vM()z^W%HF5$\"3CW!QMXc*\\HF17$$!31:o&)R[,L;F5
$\"3ASWq\"*f\\[HF17$$!3jyXz))=K>HFeq$\"3;Y1A>\">j%HF17$$\"3%=.*>qd&*f5
F5$\"34([SJ)HSVHF17$$\"3%RTxfAVCR#F5$\"3N-&*RY)e(RHF17$$\"3%[hfN\\T6o$
F5$\"3R$3'*f%QSNHF17$$\"3Ng*[qOjC!\\F5$\"3ZB`&*H\"o.$HF17$$\"3eNp.6Vx%
*fF5$\"3#*)>Yc#4\"\\#HF17$$\"3!zKh:Zv^)pF5$\"3yrsPi%4*=HF17$$\"3q*)\\&
oDIo&yF5$\"3UXn]-9U7HF17$$\"3C/x>O2#\\f)F5$\"3m]IU!o7b!HF17$$\"3q]&oa#
poa&*F5$\"3[;Y!)R58#*GF17$$\"3agOKZHU!)**F5$\"3_-bHn'*3yGF17$$\"3m)y?P
+\"p****F5$\"3#*HV'3ikQ(GF17$$\"39I9u=lNp**F5$\"3wt7AM6kpGF17$$\"3?F(*
yj*p&*))*F5$\"3!)*3!*)oJVlGF17$$\"3uoUo7rsg(*F5$\"3'o]-%4YDhGF17$$\"3e
f25V\"p'e$*F5$\"3#e%3h[//`GF17$$\"3I'pEC?]6x)F5$\"33oe(=f0^%GF17$$\"3I
/$*)p>coQ(F5$\"3*)HwoG2eKGF17$$\"3#o9%oRzl$e&F5$\"3$*[7&['[e@GF17$$\"3
qR(=5vj=Q%F5$\"3ME;T-m.;GF17$$\"3i#p!)HbWd4$F5$\"3\"G!H]o-=6GF17$$\"3y
oPnZ00]<F5$\"3MXr'R)e02GF17$$\"38Xb[0Sy1PFeq$\"3(=(>%RQ*o.GF17$$!3Ks])
eE\\5d)Feq$\"35c#e#[7N,GF17$$!3ij3'H8g>2#F5$\"3+*pR8B9'*z#F17$$!3[=!*Q
!)HcbKF5$\"3g=RjguY)z#F17$$!3_C&p[ig+R%F5$\"3E;Xig6*yz#F17$$!3=TT15S*z
b&F5$\"3-ONMC7)yz#F17$$!3GLP[(pP]i'F5$\"3y6/B'*[Z)z#F17$$!3#fp`*>D#=d(
F5$\"3OP,\"*\\Ch*z#F17$$!3K1wkqA;\"Q)F5$\"3?u%ojGC7!GF17$$!3s>x&\\`^<
\\*F5$\"38<(3qL=`!GF17$$!35u&4gD/*z**F5$\"3ZM+S#*zB5GF17$$!3=IW*G'fe\"
z*F5$\"3e$\\4r#z`:GF17$$!3+)oYF!Hd/*)F5$\"3Kopfo\"e-#GF17$$!3A[z%HX<!G
vF5$\"3%HNo+dkM#GF17$$!3oFpUb#R!)p&F5$\"3UfpA'3&3DGF17$$!35Y#)\\.,pgXF
5$\"3L^C'=vd^#GF17$$!3y-icQ_:VLF5$\"3N^#[+a/Y#GF17$$!3OhGvn7%o1#F5$\"3
Vr'Gp^)QBGF17$$!3[&)zK![#)=a(Feq$\"3UDp9-W[@GF17$$\"3T$*Rp]%3,B'Feq$\"
3q2V6\")Rw=GF17$$\"3IW!f=#HR))>F5$\"3,#G3-L'G:GF17$$\"3@oWWq*fgJ$F5$\"
3PcVGD^16GF17$$\"3_A\"\\AMH3e%F5$\"3KC#Gc9Gh!GF17$$\"3I5z<G+CilF5$\"3g
&fxw3bfz#F17$$\"3W+2L$z\"en\")F5$\"3/VnNAB0%y#F17$$\"3]!R$)f$\\[t$*F5$
\"3=Dh6TaspFF17$$\"39\"Ga$*H&H`**F5$\"3=c/gOxYaFF17$$\"3->1Mh_d&***F5$
\"3Ws=GitX]FF17$$\"3;]Qi7h3$***F5$\"3'*Rf,f;WYFF17$$\"31+W$=+Re%**F5$
\"3t:>j;EVUFF17$$\"3'Gf>PaXS&)*F5$\"3x#oD^>U%QFF17$$\"3V1zm%zh'Q&*F5$
\"30Q:`%ek0t#F17$$\"3'GP5)=)yD0*F5$\"3\\e$3i)=!Hs#F17$$\"3]\"Qz\\b#fVw
F5$\"3M7ABxf')3FF17$$\"3&p[LtTaJr&F5$\"3'Q$*yIf*e'p#F17$$\"3y>hV=w4IYF
5$\"3u3.J\"HI9p#F17$$\"3)oc)=-0zsMF5$\"3MS7zZ&Roo#F17$$\"3-j\\Rj@zfAF5
$\"3wV+,R.&Go#F17$$\"36'pDiya0,\"F5$\"3y$zu\\*f[zEF17$$!3aX@g;\"Gi%RFe
q$\"3qmD8o$)\\wEF17$$!3;*4?K0,?z\"F5$\"3#=qR')o+Vn#F17$$!3mW[%[/iR:$F5
$\"3s!=r^x!)Gn#F17$$!3I\\@C#)ye`WF5$\"3UHIdu5@sEF17$$!3c[x$od()pa&F5$
\"3'Gj\\Nf9An#F17$$!3A'G=h$\\%=b'F5$\"3c&>G&o$\\Fn#F17$$!3W_?'y=?@X(F5
$\"39Rw[-lwtEF17$$!3-VwKCHWL#)F5$\"3]!3y))R4_n#F17$$!3-o35!*)y)[%*F5$
\"3[b!*QeeTzEF17$$!3/#3NmW+)z**F5$\"3e;55F5d%o#F17$$!3.uJ+RZ_O)*F5$\"3
oAqlCDU*o#F17$$!3'z.uT*\\\\2\"*F5$\"3%yB-j3XQp#F17$$!3K&=Dt*RN#p(F5$\"
3-S55,j`(p#F17$$!3(H;>WsZ-t&F5$\"3!o0p(o,T*p#F17$$!3Y$\\oRowkl%F5$\"35
]bWU4^*p#F17$$!3UO![)=634NF5$\"3QgI==?0*p#F17$$!3+<D!)RF?1BF5$\"3+Xu-j
3+)p#F17$$!3GTq#z]go1\"F5$\"3s:3\"zIMjp#F17$$\"3K78Tjb?jCFeq$\"3Mjpb!)
*>Rp#F17$$\"3o]-TmnCb:F5$\"3Af#*)p\"\\\"3p#F17$$\"3R0Cs$>3t$GF5$\"3#zK
))\\JFqo#F17$$\"3;%\\yC'*e.2%F5$\"3kkBEird#o#F17$$\"3-_BSrZ13_F5$\"3Eo
A4\"*ehxEF17$$\"3+\\WVYMxfiF5$\"3i%pEXx!4sEF17$$\"3#y4:s&)=\"3sF5$\"3g
LK-z00mEF17$$\"3KO8tQ;WP!)F5$\"3m6!yhd_&fEF17$$\"3K'y'[)G50L*F5$\"34*[
3X^eZk#F17$$\"3'z&o^J$z7&**F5$\"3#pZdWhF*GEF17$$\"3kY'f,7YU***F5$\"3Uh
AC/T/DEF17$$\"31Jzo\"QD_***F5$\"3qa0=%*[:@EF17$$\"3yWv0-I@a**F5$\"3Y^T
Xw3F<EF17$$\"330![7G\"Qr)*F5$\"3)>'4V@HS8EF17$$\"3uPY&[+E=e*F5$\"3js.<
)Hedg#F17$$\"3?,^oD.UJ\"*F5$\"3QW8GCeI)f#F17$$\"3ROI;uf&Qu(F5$\"3u7$pg
E**Re#F17$$\"3([<O-R6\\\"eF5$\"3y!=()G0i9d#F17$$\"3Mcml:C-)p%F5$\"3MEF
RAM1mDF17$$\"3)oQD=#*=+]$F5$\"3qmSzdWFhDF17$$\"3Y27WX_eTAF5$\"3W,4FPC8
dDF17$$\"3!))z.)44\\W%*Feq$\"370!y!pMm`DF17$$!3I*p'\\*z3Si#Feq$\"3)GpX
i'=3^DF17$$!3ss>R]XUl9F5$\"3)eDJ\"QL3\\DF17$$!3Kq()zi&zqk#F5$\"3WTRv)y
hwa#F17$$!3oZ.-b^8!z$F5$\"3uNJp3F!oa#F17$$!302t!QOt7.&F5$\"3%)*>]&*=![
YDF17$$!3wG9C?t2whF5$\"3_Bc[c&>oa#F17$$!3FYd,eti-sF5$\"3q$RF/Ghxa#F17$
$!316_WH!o74)F5$\"3Es'=*)HM#\\DF17$$!3M3%e!=\"H^G*F5$\"3m$z&z+K$Hb#F17
$$!3/!*4s=757**F5$\"35D')\\o)Qvb#F17$$!2GU@R0)******F1$\"3_tx1^X;gDF17
$$!3avu.HUu8**F5$\"3'pbF#o1ziDF17$$!3-.`Tg=%[l*F5$\"3Kqxa)[D`c#F17$$!3
0Q;x](*zF#*F5$\"3!z+R0()ywc#F17$$!3kxPLjQtqzF5$\"3A_;Wl]OrDF17$$!37%y
\"fA.)p?'F5$\"3um85Wd^tDF17$$!3/FELj>8\\]F5$\"3g$o?\\`tQd#F17$$!3!p@-z
_')>z$F5$\"3];8!eJ_Nd#F17$$!3CZpJ\\sEgCF5$\"3'>Ab/?.Dd#F17$$!3s$Q-B%R;
!3\"F5$\"3g`O_<;pqDF17$$\"3!e7(\\X_fI8Feq$\"39*)\\$RV\"\\oDF17$$\"3Cr2
<b+KW8F5$\"3GHE#*47qlDF17$$\"3l(pV2v[d`#F5$\"3m0Ri\\iKiDF17$$\"3)G`gzF
p(*o$F5$\"39l4R$[!QeDF17$$\"3t=)pZ0XF$\\F5$\"3kCJ=[!RKb#F17$$\"3A8()zN
#4?3'F5$\"3Qv.X$eOua#F17$$\"3]G#3?EGd6(F5$\"3g<O_64.TDF17$$\"3yXj$\\oj
U,)F5$\"3ah!zVJ\"4MDF17$$\"39g-Cw0I]#*F5$\"3***Q>ZcM._#F17$$\"39ze\"Q=
'f/**F5$\"3J_9`q@i0DF17$$\"3swmB?\"pe(**F5$\"3+9$G\">?^,DF17$$\"2-FrOO
(******F1$\"3omSEV()Q(\\#F17$$\"3S@&*4()o'o(**F5$\"3cod8I`E$\\#F17$$\"
3'\\%=A-qe1**F5$\"3)4<*enZ:*[#F17$$\"32yWO&zGhi*F5$\"3!y1\")3yB5[#F17$
$\"3%*Rg7>\"=R;*F5$\"3=%*H'fP(4tCF17$$\"3#yf3z,#3mzF5$\"3o$4$)pn#=gCF1
7$$\"3**QOf0pt@jF5$\"37M$e')GB'[CF17$$\"3wm?9G8C?_F5$\"3%)>>$zY'zUCF17
$$\"3%4)p\")[UcBSF5$\"3*=(e87<fPCF17$$\"3cfQwt[_`FF5$\"3&GQzBkaIV#F17$
$\"3y(\\&pN+GL9F5$\"3mDWu0(=#HCF17$$\"3(>Vh%346H8Feq$\"3K;h77')>ECF17$
$!396Vf`#>(p6F5$\"3@*\\FN.fQU#F17$$!3GU40_vW_CF5$\"3#y&o1e`>ACF17$$!3+
9)*))z*[Mp$F5$\"3=P2zt9>@CF17$$!3iF))\\\\R'3([F5$\"3LB#Hn3?3U#F17$$!39
%45!)G7b'fF5$\"3,C]I2B/@CF17$$!3w'o_h_$zepF5$\"3\")G01G+\"=U#F17$$!3$Q
;\"QV'HQ$yF5$\"3]hqHse1BCF17$$!3uFBdc\\7o\"*F5$\"34lQifgvECF17$$!3wT)y
#pFX'))*F5$\"3sV]K'Q>:V#F17$$!3wzNx^I%4'**F5$\"355oZ5(\\FV#F17$$!3].q/
6[i'***F5$\"3^8)y[z$*RV#F17$$!3)ya'\\G!fL***F5$\"3vZj'z#>CNCF17$$!3Q^p
$yUe6&**F5$\"3gZ2S#Q%[OCF17$$!3o*f&>!\\h2v*F5$\"3)Qz*Q%\\8*QCF17$$!3+1
ib(*Rb)R*F5$\"3$)eMYLW?TCF17$$!3!Rvkwkss9)F5$\"3m\\[*QMO`W#F17$$!3o(RL
/L!)eI'F5$\"3szm#yJ)yZCF17$$!3=6zD_)H\"e_F5$\"3!f32h\"[>[CF17$$!3nQAO;
k[BTF5$\"3I(GqB0R![CF17$$!3Y4x>:0q?HF5$\"3y,o*R3\"GZCF17$$!3+0L1E(['p;
F5$\"3yv#p+#4*eW#F17$$!3dSaW:,tNKFeq$\"3#)y!Q%4PsVCF17$$\"3$\\d&*enC%G
5F5$\"3mxSg4[#3W#F17$$\"3'[[&oE.ghBF5$\"34J#H*=x>PCF17$$\"3)[le'3Wb^OF
5$\"3**z%))HEfGV#F17$$\"3Y\"3\"H%>jAw%F5$\"3`K=?VJMGCF17$$\"3eK'=ulH:!
eF5$\"3O^X\"zO*HBCF17$$\"3m%R6N(Hw`nF5$\"37W%33$ow<CF17$$\"3AJbHozn/wF
5$\"3(G5_yv\"z6CF17$$\"3=>(er2f]1*F5$\"3iyo;gkG(R#F17$$\"3#G(Hw6OCk)*F
5$\"3[(\\j#)Q\\:Q#F17$$\"3m3XtZq%*\\**F5$\"3)3'R4q/oxBF17$$\"3_\"ewVG'
z$***F5$\"3)ov(>QZztBF17$$\"3u.**)4(og&***F5$\"3/vaV!3.*pBF17$$\"3Q3zQ
\">r`&**F5$\"3;6Zf4k,mBF17$$\"3CtWf*=9'\\(*F5$\"3'y>D\"H:IeBF17$$\"3'H
hMYo$)*z$*F5$\"3+!*3,dht]BF17$$\"3A/A`vxB!=)F5$\"3W;>&>]<kL#F17$$\"3j
\\#[1XqaV'F5$\"3@kE5]<jBBF17$$\"3o/!R94?_M&F5$\"3\")RE9-#QxJ#F17$$\"3[
#=)\\q(Ru:%F5$\"3Fkye)[hCJ#F17$$\"3kk3lm>!Q*GF5$\"35pHZe&[yI#F17$$\"3y
kYipKOx:F5$\"3iEjI\\U$RI#F17$$\"3'*pdJyiV)Q$Feq$\"3p;eMHj'4I#F17$$!35B
&)=U)>#\\!*Feq$\"3w\"3pHL<')H#F17$$!30d(RAhxY8#F5$\"3Y\">!*\\2&)oH#F17
$$!34DjB/UQJLF5$\"3'pp\"*=#yv&H#F17$$!3H]zP`oO.XF5$\"39#>.AM3_H#F17$$!
3ttek\"yEAg&F5$\"3ES&y(o(Q_H#F17$$!3OhBP@975mF5$\"3UPkP)>2eH#F17$$!3h]
\\lR_o5vF5$\"3gQ\\,#*H'oH#F17$$!3#pK)G%zu%\\!*F5$\"3!*z12'[>1I#F17$$!3
Y#oqs*z\\v)*F5$\"3(Rt_Xg;dI#F17$$!3%)HQap'Rc&**F5$\"3a-F3h1(pI#F17$$!3
w'*zW%4_`***F5$\"3p`p*3+S#3BF17$$!3E#GSA+uW***F5$\"3`-+q#H9&4BF17$$!3o
T(*ze*3I&**F5$\"3$yj'3&=$y5BF17$$!3-=Gj.d:\\(*F5$\"38;2d<NE8BF17$$!3yl
X&)4.5(Q*F5$\"3)oV4(y$*f:BF17$$!3)zC812K7=)F5$\"3)))QD;?)f>BF17$$!3DiS
&pj\\BU'F5$\"3PM\"QE@U?K#F17$$!3yBnX)>[FN&F5$\"3**4;qpu^ABF17$$!3K!GVq
W+#*=%F5$\"30]\")fF(*RABF17$$!3/k$R\"fy7_HF5$\"3KtRQ:Xk@BF17$$!3)RGD([
BCj;F5$\"3<90Q]&=-K#F17$$!3c[<0^B4AXFeq$\"3-cc5(\\(H=BF17$$\"33Za/bIJb
wFeq$\"3ZC#4yu$y:BF17$$\"3'>X(HYq\">(>F5$\"3rxTfO%yEJ#F17$$\"3eGoP\"=a
!\\JF5$\"3In%\\>Z\"**3BF17$$\"3op.4#Q,?N%F5$\"3XZ)G!G=W/BF17$$\"31O\"G
;17;[&F5$\"3/q\"G`J&G*H#F17$$\"3f1Q\"\\!4&)=lF5$\"3)e--n/lNH#F17$$\"3#
HLgZ<RiW(F5$\"31P'y\\jLtG#F17$$\"3_E4-9g2-*)F5$\"3W4F7z)*ptAF17$$\"3/7
qP=#4%p(*F5$\"3%eFR#z9')eAF17$$\"3EW!G1Y$H$*)*F5$\"3g'**>p&*pZD#F17$$
\"3Q*)oV+N;q**F5$\"3G;r()R2l]AF17$$\"3C\"p\"\\TSl****F5$\"3[Hp.]o^YAF1
7$$\"3eV*\\4%\\i\")**F5$\"3f\\F0z8QUAF17$$\"3Ev8zLfd.)*F5$\"3cRdeN!eTB
#F17$$\"3o0)>_!pRR%*F5$\"3w`KhuW3EAF17$$\"3i=20**)zcO)F5$\"3ap%fK32D@#
F17$$\"3)\\nT)))f())z'F5$\"3R^i'*p8A+AF17$$\"3zu)4!)RTdq&F5$\"3W>mh)3'
*Q>#F17$$\"3N$HM]o6.]%F5$\"3c>[\\Ft@)=#F17$$\"3/t(H?+6j?$F5$\"3g)ysWeT
K=#F17$$\"39H.)Hn1#\\=F5$\"3W\"zq7_6!z@F17$$\"3e<[akF@DfFeq$\"3+qTp^y&
e<#F17$$!3SnpfIsjOnFeq$\"3?+SwbVMt@F17$$!3=H-a-#[!H>F5$\"3Y9]P.2Zr@F17
$$!3O%30%)>0N:$F5$\"3W(HChIE-<#F17$$!3?9wfjiM:VF5$\"3)[z,k6%fp@F17$$!3
o]&ys-_%4aF5$\"3;=qY7$G&p@F17$$!3m'G')H**['=kF5$\"3+Wh'4d!**p@F17$$!3z
!QcZU'4FtF5$\"3_Z$y)zU$4<#F17$$!3UM^$ya5^())F5$\"3Y%4K]2KV<#F17$$!3Oxs
>ar%=y*F5$\"3'oP0%[;1z@F17$$!3SNM_(GQ***)*F5$\"3C'oSy9t.=#F17$$!3K_&p7
J-F(**F5$\"3#4v?/\\5<=#F17$$!3ygKdtg!)****F5$\"3S\\)3eNhI=#F17$$!3;\\f
>za7\")**F5$\"3c^PvhLT%=#F17$$!3=7*G=dip!)*F5$\"3Im]\"pHrq=#F17$$!3yK`
tu&*R`%*F5$\"3CM)QC:'e*=#F17$$!3s(*eJM\\,t$)F5$\"3[u\\;<i[$>#F17$$!3]=
^O?sA(y'F5$\"3'QkH(3i1'>#F17$$!3=F\"))))pL:x&F5$\"3)*)=`g=Sn>#F17$$!3#
\\8G&HFEdYF5$\"3!Gp$yRi&o>#F17$$!3M!)[J$*fWjMF5$\"3oD#oArnj>#F17$$!3]'
)Q=3QZ5AF5$\"3m9^1s#Q_>#F17$$!3#eF:a/X7^*Feq$\"3-K_utP\\$>#F17$$\"3HG^
X'*[(pB$Feq$\"3!>/OR^,6>#F17$$\"3!zfA@k`Kf\"F5$\"3'H+dO))e!)=#F17$$\"3
P6ZA%3\"*o$GF5$\"3?Tc,aQP%=#F17$$\"3YK*H_qbA8%F5$\"3!>$eBk_nz@F17$$\"3
w,Ie#=Z'[`F5$\"3aAlIR0Gu@F17$$\"3E-GM'e=GY'F5$\"3)\\Joq*)R#o@F17$$\"31
)\\%4alZ`uF5$\"3ye$Q(**ehh@F17$$\"3!yQ,=fn=#))F5$\"3g]Ko^i**[@F17$$\"3
)3ueP*>%po*F5$\"3/%[:%[>IN@F17$$\"3WC%Rg`(fM)*F5$\"3F%Rb)p<CJ@F17$$\"3
A\\7yA'*yN**F5$\"3[6\"H?OZr7#F17$$\"3#Gp/I<S+***F5$\"3<[k*GdJI7#F17$$
\"3+L(R0)G4(***F5$\"3$z(HV^t!*=@F17$$\"3m9M\"*GYpp)*F5$\"38.FCXbo5@F17
$$\"3S`:A\"f**fb*F5$\"3KDTXG_e-@F17$$\"3!pD^RN\">&[)F5$\"3a9)GB:N!)3#F
17$$\"3q4ieMi\\]oF5$\"3U]%40r\"*[2#F17$$\"3MyM24ZZkeF5$\"3uiUme>2p?F17
$$\"3+n6;R>\\#y%F5$\"3+8%[VO#yj?F17$$\"3.;^wkDDAOF5$\"3=_AQ!fn!f?F17$$
\"3^EE*)\\<u-CF5$\"3;\\zr$*G'\\0#F17$$\"3u-Au2H!H9\"F5$\"3#ok+i/\"\\^?
F17$$!3a\"Q5xWrmN\"Feq$\"3W@&H\"p/n[?F17$$!3\")y>sFP,79F5$\"3q&HOyR0l/
#F17$$!3U^y'>g=_m#F5$\"3C`$oWK*)\\/#F17$$!31:T-#Qv#)*QF5$\"3@iX/HP4W?F
17$$!3!)=hZ4N([1&F5$\"3OhNb8h#Q/#F17$$!3_@:\"RcC^9'F5$\"3HJ#e$*[YT/#F1
7$$!3B5Dt)=717(F5$\"3$40<O%[+X?F17$$!3?\\M6oyDp()F5$\"3MXQ)e`C$[?F17$$
!3#*4M9aaM\\(*F5$\"3[s\\SA*4J0#F17$$!3iiqFvd(3()*F5$\"3!)H%HVVKV0#F17$
$!35'zJVsBE&**F5$\"3uvEHp2eb?F17$$!3Mv&eV$)fU***F5$\"371s4yZ%o0#F17$$!
3K`]E+jh&***F5$\"3a#ev$[U6e?F17$$!3;h,<#zIw()*F5$\"33<_2b'G11#F17$$!3@
^b6Ttc+'*F5$\"3AC8(>fUI1#F17$$!3,`+Q'pQig)F5$\"3-?aq$z0s1#F17$$!3o.D'o
6#ytqF5$\"3Y\\9?H`4q?F17$$!3)f$)zL7;g1'F5$\"3AXFE%4U42#F17$$!3%zs4(pk[
\\\\F5$\"3OSG4\"o=72#F17$$!3TZM!4`7Uu$F5$\"3)\\j_p2s32#F17$$!3/zW\"p'[
!=Z#F5$\"3_6i\"HWg)p?F17$$!3=$3vEIvz:\"F5$\"3wj2\")[$e\"o?F17$$\"3=<>L
PfEl<Feq$\"3uuw/tpul?F17$$\"3&pTYM6xy]\"F5$\"3]!4GJi@E1#F17$$\"3pq[S1)
3B\"GF5$\"3mUm^_.ze?F17$$\"32k9<!R>5%RF5$\"3aHsrOrwa?F17$$\"3Vg;FmT.8]
F5$\"3#p`K.0:-0#F17$$\"3+yCzZ5$H,'F5$\"3#\\%)=%\\];X?F17$$\"3b*)=RTaKE
pF5$\"3M%4pIDb'R?F17$$\"3!o'eUH&e:n)F5$\"3`BmQ%\\Q]-#F17$$\"3th<D#3g/s
*F5$\"3?mn*)G?#)3?F17$$\"3n6#=@!*QG%)*F5$\"3i,[VtXI0?F17$$\"38=y8+pSI*
*F5$\"3@w&3f8i<+#F17$$\"3o\\YlxV&G)**F5$\"3#)GDhPI?)*>F17$$\"2YF@b%e**
****F1$\"3')fYwecj%*>F17$$\"3'*G<6!oq<)**F5$\"37.&o)3%o5*>F17$$\"3DDps
PLCG**F5$\"3CM:(pp4v)>F17$$\"35jVjEJgR)*F5$\"39)=D>!z'R)>F17$$\"3G+#[v
`jhr*F5$\"3!)o;vU8X!)>F17$$\"3))GyGV`f#p)F5$\"3qZ**3MUdk>F17$$\"3gMx0w
fE+qF5$\"3kWDI[%=-&>F17$$\"3!\\imkP[J*fF5$\"3<SNQ'y1T%>F17$$\"3=uaO%)z
]!)[F5$\"3Ljmo<*e&Q>F17$$\"3'yFVWKN>o$F5$\"3d0<ijdiL>F17$$\"3L5)=8&R`=
CF5$\"3Cg8<AuMH>F17$$\"3Gh&4hw\"z`7F5$\"3IuQvoU5E>F17$$\"3l<`a$zpW9(!#
?$\"3!>')3eL=M#>F17$$!3JORiZb!>6\"F5$\"34v,1TSH@>F17$$!3)o4h!zJkzAF5$
\"363Q4;zs>>F17$$!3WZ\\fT\"[Qb$F5$\"3(>l.l5>'=>F17$$!3')3)f(*\\h\\w%F5
$\"3BvSzhS<=>F17$$!3U)=t4J#[\"*eF5$\"3*o35$RLN=>F17$$!3.b3V29T8pF5$\"3
))G*zFT1\">>F17$$!3wNGC%y!>#e)F5$\"3F``L%G2@#>F17$$!3)R`4(RL@O'*F5$\"3
+wfi])yl#>F17$$!3$oeV%**eL1**F5$\"3n*R!p:h8H>F17$$!3?#z:$[D%*****F5$\"
3cJz?.2zJ>F17$$!33.B`d`O:**F5$\"3<$QBnqZW$>F17$$!3=&R6YO6Tl*F5$\"3)39K
n<7q$>F17$$!3/wDKc9??()F5$\"3C/=@gw?T>F17$$!3!fNt!Gg)RD(F5$\"3&3OU*)*z
?W>F17$$!3eAm,pM79jF5$\"3]o&Qa#z7X>F17$$!3/C?'*fScq_F5$\"3a)[@^WQb%>F1
7$$!3kg%Gr<Y/9%F5$\"3]T*zUp!RX>F17$$!3*[N`luHB%HF5$\"3)[fzw(\\kW>F17$$
!3mF')=F1l_;F5$\"33s)**R28K%>F17$$!31+f_;\\WRLFeq$\"3F5*>WF)3T>F17$$\"
3E@!yv;qi!**Feq$\"3%R2m+Jh#Q>F17$$\"3SD8$eB*z(H#F5$\"3vqv#[CNZ$>F17$$
\"3W&)>)f%)Q1c$F5$\"3=WFV*)*R0$>F17$$\"3C/,U%fR8w%F5$\"38$Q[)>CpD>F17$
$\"3;%e*GEt%*yeF5$\"3O&zI;*4B?>F17$$\"3-$)\\@$*z&R*oF5$\"3:8X:t[?9>F17
$$\"3c0*=Y\")pke)F5$\"3>[cs&y#H+>F17$$\"3c+1C%)zwZ'*F5$\"3/T'4>\\t[)=F
17$$\"3)*GPZcm)H*)*F5$\"3Umf=u8zx=F17$$\"3#f\")RPg[k***F5$\"3!Qo:(Hciq
=F17$$\"3Mt&4^RxW***F5$\"3)=22wtKq'=F17$$\"398-b=8nc**F5$\"3%3dk7hWM'=
F17$$\"3QN!)\\Ef;$))*F5$\"3%[%yG@)p)f=F17$$\"3ibN5qZAu(*F5$\"3ef%f/*oJ
c=F17$$\"3-EG<&y@!e))F5$\"3fA:7#f73%=F17$$\"3xxOM$G0:I(F5$\"3#H5e&4,lE
=F17$$\"3X\"=\"RR4R7jF5$\"3BBs:#=]-#=F17$$\"3z*\\zH6Sx?&F5$\"32o#[B3=W
\"=F17$$\"3`9Rim8x2SF5$\"3ym=p$*4@4=F17$$\"3pmSnyyWMFF5$\"3/%>\"4gYn/=
F17$$\"3!e#=-.Otf9F5$\"3#z:PL6p4!=F17$$\"3.Iw1@>/-;Feq$\"35y#pOFOzz\"F
17$$!3:L(*>`'[?9\"F5$\"31Ky_PLe&z\"F17$$!3Nv%oU$p)[U#F5$\"3!4k\\J%f!Rz
\"F17$$!3UD\"G4)*zYb$F5$\"3KLuq&)[&Hz\"F17$$!3D(H,,zGYj%F5$\"395(Q<3GD
z\"F17$$!3[8g=j,f\\cF5$\"3K0L$yG)f#z\"F17$$!3%>q?[;K`e'F5$\"3)*oRV\"HJ
Jz\"F17$$!31_5luL?=%)F5$\"3'f&Hamu+'z\"F17$$!3M,b*4e&3(e*F5$\"3Ktq&ee'
e+=F17$$!31G-cfJ]d)*F5$\"3(QSs;cYG!=F17$$!3;.D1n&ow)**F5$\"37$[%zrA?0=
F17$$!3i,/>%4*\\****F5$\"3S!fmo'\\R1=F17$$!3[%e`^\")Hd(**F5$\"3G@OgOie
2=F17$$!3!)oMO_`W;**F5$\"3701y!fn(3=F17$$!3>r1$zwc=#)*F5$\"3WTNZu0$*4=
F17$$!3uSy<nDbI*)F5$\"3IS'>>be[\"=F17$$!3'o!o@**R6btF5$\"3oe(HGy2%==F1
7$$!3]L_z^&G\"pkF5$\"3RB#oS+_$>=F17$$!3'yI[`;mb[&F5$\"3$>WU29O)>=F17$$
!3rg)z&yBE>WF5$\"3zg_e<e\")>=F17$$!3;i+/e2I'G$F5$\"3r+oLoUD>=F17$$!3=
\\GxC^*o)>F5$\"3RUxSw<)z\"=F17$$!31#*46r,B9lFeq$\"3NR?stw*f\"=F17$$\"3
1G,\\s$Q(epFeq$\"3/=\"Gh-*G8=F17$$\"36D>ql!R0.#F5$\"3z,Fa^g&)4=F17$$\"
3-Mk$y>[vE$F5$\"3:/-Ite#f!=F17$$\"3))yK%[In2X%F5$\"33$QyOOt8!=F17$$\"3
s/\"><O=2c&F5$\"3OT*pS>Jiz\"F17$$\"3)olzTZH\"zlF5$\"3mAlyU<a!z\"F17$$
\"34o(G$Gv3Q$)F5$\"3-xG?t:9x<F17$$\"3$Rk/Pfn?]*F5$\"3y=IO'pl@w\"F17$$
\"3M6]CJik7)*F5$\"3w+j!\\2Z\\v\"F17$$\"3=]$4>:2\\(**F5$\"3'fdF'o>hZ<F1
7$$\"3o%>Q>Aj&****F5$\"3MH$30sBRu\"F17$$\"3e)G2!*y(R')**F5$\"32&yf$eSB
S<F17$$\"3dfw$*[1YN**F5$\"3_Kki%G_lt\"F17$$\"37)oj?YWp%)*F5$\"3C%G;>n(
)Gt\"F17$$\"3'>$QoLi2X!*F5$\"3HIr])=$\\<<F17$$\"3K&Gx`j%))4wF5$\"3:aX'
G5$H.<F17$$\"3%Rw.zaOjk'F5$\"3mI-B/.g'p\"F17$$\"3q8#\\?XBnb&F5$\"31/N!
eQu/p\"F17$$\"3!QAQeo7<O%F5$\"3)o3,ZDz\\o\"F17$$\"33s$QusqR3$F5$\"3m%G
AN&o;!o\"F17$$\"3OU74=n=B>F5$\"3)G(oc)>kln\"F17$$\"3i_(4V/)\\ZtFeq$\"3
3OAjaf_t;F17$$!3S-@\"R[=Dk%Feq$\"3w!>\"o.81r;F17$$!3Kr:7o\"yll\"F5$\"3
U^7;!=r\"p;F17$$!3h[03B91<HF5$\"3os+8\"\\mxm\"F17$$!3@+F3Y!H'GTF5$\"3]
Xa!>Q.qm\"F17$$!3yrb$o\\k4F&F5$\"37iU=*=^om\"F17$$!3\\+-zv@\"\\K'F5$\"
3mjTK0!psm\"F17$$!3MT)3VD\\([\")F5$\"3'>\"3;r7rp;F17$$!3%QO>u$=*=R*F5$
\"33a(Gax0Qn\"F17$$!3My1gXC#*f(*F5$\"3GV/Jy,@w;F17$$!3!op9f!)z2'**F5$
\"3=%)4n3Qwy;F17$$!3[N*>**eHt***F5$\"33rN?`*o+o\"F17$$!3=s()\\CN-\"***
F5$\"33(Hh,1y8o\"F17$$!3\\!)*[?k))=%**F5$\"3uR>W.*zEo\"F17$$!3Q]#H0dN,
&)*F5$\"3S&p!3)GjRo\"F17$$!3R\\usW#>I5*F5$\"3A?tV8Kb)o\"F17$$!3;$Qc.cB
Vx(F5$\"3]g#QRG&3#p\"F17$$!3$)ytTzLX:oF5$\"3ivN3TOQ$p\"F17$$!3Y6P<T?)=
s&F5$\"3Ctm<$[.Tp\"F17$$!36sCnvHA:XF5$\"33chwm\\<%p\"F17$$!3d1#3xoC$>K
F5$\"3q[\"4E1TNp\"F17$$!3uR(4FWj)**>F5$\"3-<`+')RL#p\"F17$$!3i!fln,*z%
[(Feq$\"3efP@=8]!p\"F17$$\"3czp3O0^[^Feq$\"3].\")*=&>.)o\"F17$$\"3?=W*
Q$Q'*p<F5$\"3E(GbG%\\#\\o\"F17$$\"3qSS#p(fQyIF5$\"3/JK@$H94o\"F17$$\"3
%))Rt&R2!3L%F5$\"3%f,!>M7@w;F17$$\"3/)R!e-?U/bF5$\"3%[p*y8A&3n\"F17$$
\"3#f;-e>(*yd'F5$\"3!ym7Wh&)[m\"F17$$\"3'f6#>0?K1#)F5$\"3#oh)H*4BFl\"F
17$$\"3K_OLD&pgM*F5$\"3Izq%)*R*>R;F17$$\"3r?(32s]ot*F5$\"3C!ezK]89j\"F
17$$\"3yGE]o4x`**F5$\"3-7=FL[YB;F17$$\"3+X>:z#*p&***F5$\"3a)Hz,#)e%>;F
17$$\"3KQ$Ql%p&H***F5$\"39@\\&\\cZah\"F17$$\"3OR(3hAcb%**F5$\"3O\\Gl<I
W6;F17$$\"3kKR.^*3P&)*F5$\"3,R'\\$)4dug\"F17$$\"3I!)=+3=)*=\"*F5$\"3_N
G*pNCGf\"F17$$\"3!QXR0'e!)4yF5$\"3!pcF0yD#z:F17$$\"3F_xLH*zK#pF5$\"3'
\\L,U>9Fd\"F17$$\"3\"f@*[3P;<fF5$\"3)*)QBE=,nc\"F17$$\"3L\\>'Q\\O)3[F5
$\"37C,Wu^Ch:F17$$\"3L;?*y8Vuh$F5$\"3#RB=o(\\Rc:F17$$\"3)*)3_f'R2#R#F5
$\"3/,'f`bvAb\"F17$$\"3l$oB$[h?F6F5$\"3UI(p3X(z[:F17$$!3Wp8zmVui:Feq$
\"3)*pj#*Gc(fa\"F17$$!3PwA7rX<P9F5$\"3W@AP=U\"Qa\"F17$$!3gS-p;7Q_FF5$
\"3d7S&[^^Aa\"F17$$!3ng+\"\\g#y<SF5$\"3Qo!*=oYQT:F17$$!3M\"oY](3[5_F5$
\"3!ft)=W1=T:F17$$!36qzc`P*)3jF5$\"3I!)om9WfT:F17$$!3uv;!*\\+U(3)F5$\"
3z9Z[hl\"Ra\"F17$$!3zFbId.<D$*F5$\"3%)QW_T=\"ya\"F17$$!3M=)oX[S2s*F5$
\"3O<0fw!=-b\"F17$$!33%He1=lc%**F5$\"3;^A)y4*y_:F17$$!3=^071By#***F5$
\"31Y')p?y5a:F17$$!3!QV*pUf*f***F5$\"3Y))QR>KVb:F17$$!3'=N=7(>Hb**F5$
\"3U:iOXOvc:F17$$!37n**QD#\\3()*F5$\"3!=ih=Zd!e:F17$$!3\"*\\X)RT1]6*F5
$\"3Gtt]e<%Gc\"F17$$!3%*y]NLRPKxF5$\"3Al]%Qt%\\m:F17$$!3*yoRN$Rm6pF5$
\"3Q%>kVJ@wc\"F17$$!3w5O*y\")o,*fF5$\"3UQ&pTOE$o:F17$$!3!3yYesD8)\\F5$
\"3_%R2SOk&o:F17$$!3kZ'>`\\X)**QF5$\"3%>\"[40mHo:F17$$!3=W&[O.&R\"f#F5
$\"3)p.JAnFtc\"F17$$!3AU\\p$*oKL7F5$\"3_\"og$=!>cc\"F17$$\"3C\"e5OpeN[
\"Feq$\"3ccQ0G.:j:F17$$\"3\"RY&)*[!)>F:F5$\"3S.!o$Qe\"*f:F17$$\"3MwoMj
`XOFF5$\"3Czh1`5Qc:F17$$\"3z?UcuXy.RF5$\"33&yN\"znD_:F17$$\"36*Qss!)*H
6]F5$\"3yDto_#ova\"F17$$\"37_a\")e;.UgF5$\"3NIgLb$\\Ba\"F17$$\"3dS<;))
)RF\"zF5$\"3#*>[oG@lH:F17$$\"3L0])GFWYB*F5$\"3s'p&4O&e_^\"F17$$\"3ev8E
4(eYk*F5$\"3MXLpm9&z]\"F17$$\"3TBH\"f:&y+**F5$\"3^(G8>xv/]\"F17$$\"3=X
$Q.Kl(p**F5$\"3cTu?*G*p'\\\"F17$$\"3e#*GF<p$*)***F5$\"3%*G*efR5H\\\"F1
7$$\"3S*4%fmM=))**F5$\"3G+h(Q<>\"*[\"F17$$\"3L]'pg!zaP**F5$\"39zh'RrN`
[\"F17$$\"3#Rl52Kv%R#*F5$\"3OH%3-L%))o9F17$$\"3j@u`\"*e:=yF5$\"3q'y9(e
1j`9F17$$\"3w.4DMjd/qF5$\"37JPh5EfZ9F17$$\"3?3QK\\Im(3'F5$\"3[4q43d(>W
\"F17$$\"3+:\"==jU43&F5$\"3lP\"e`&p#oV\"F17$$\"3_iu/SlE**RF5$\"3&***oH
]k=K9F17$$\"3\"[tOl[z4q#F5$\"38*p64=wvU\"F17$$\"3C/*p^gz8N\"F5$\"3+d</
#e(pB9F17$$!3i^Zgf&f$*Q#Fj[r$\"3J%G&oE>d?9F17$$!3A&Q)>pGr)R\"F5$\"3tuc
_$=1#=9F17$$!3<e$HI%3BHEF5$\"3X]B9:fq;9F17$$!38Q>?&oZ#=QF5$\"3p:vL8[\"
eT\"F17$$!3%\\F]CM&*p%\\F5$\"3<OgQbs]:9F17$$!3cQ_J\"3dw*fF5$\"3@>lg2%[
dT\"F17$$!3/'4G7-c`%zF5$\"3l\"eb*[g'zT\"F17$$!3;t\\=Tt\"eH*F5$\"3]pS(o
U4?U\"F17$$!3))=0^\"G4om*F5$\"3aJ9D;;9C9F17$$!3[V;%G4k=!**F5$\"3yfOI%z
<kU\"F17$$!3Wp3oujn(***F5$\"3g&[\"*>es(G9F17$$!3up'p.z)*G&**F5$\"3okp%
[\\R6V\"F17$$!3NMGQT+yK$*F5$\"30^8\"HiAiV\"F17$$!3j,,gC'*4N!)F5$\"3WYF
iiFES9F17$$!3-\"[8_k.>>(F5$\"3)=]Rr*HjT9F17$$!3!)*30f*=MEiF5$\"3&*)f[
\"f.`U9F17$$!3?(3J'4G%[:&F5$\"3b:gBi^*GW\"F17$$!3_dMGxsj&*RF5$\"3[Mn9
\"ywEW\"F17$$!3eco!Q&yEpFF5$\"3\"H:pTFN=W\"F17$$!3[de48D%e\\\"F5$\"3pj
Vfh1MS9F17$$!3mJwum`**p>Feq$\"3-0Zr3S<Q9F17$$\"3pTHr#*3>06F5$\"3wZqD+y
KN9F17$$\"39f6__9%RQ#F5$\"3diaXY-#=V\"F17$$\"3G/!)G72[AOF5$\"3$)*HWDWd
wU\"F17$$\"3%y5!)R_<**z%F5$\"3Bo[Zgd'GU\"F17$$\"3'**\\-%*Q\"R'*eF5$\"3
6C]%)z@[<9F17$$\"3()*RSj#)o%4xF5$\"3%45MS>dcS\"F17$$\"3\"eU<28(HW!*F5$
\"3aN7gJ=D#R\"F17$$\"3TPX&zi?h`*F5$\"3-;#f^(*GXQ\"F17$$\"3O#4U\"*zyW&)
*F5$\"3Y!*G7zqew8F17$$\"3YpMjX@nY**F5$\"3)4o6JojDP\"F17$$\"3eW.D*f!e$*
**F5$\"3=2?%[]@&o8F17$$\"3!envmf!*\\***F5$\"3U:.O-GZk8F17$$\"3E!*z^=d*
3&**F5$\"3qviTj)H/O\"F17$$\"3;WYbal^r$*F5$\"3y'o&=5lGX8F17$$\"3)=!=vzD
Av\")F5$\"3\"zi66#>1J8F17$$\"3VE&R%f')4BtF5$\"3k4!=v=^TK\"F17$$\"3O1R)
Hv[pL'F5$\"3:D?]u*QxJ\"F17$$\"3xSx3#>@[B&F5$\"3]n$G/3$*=J\"F17$$\"3Uw
\"f)4p)o.%F5$\"3KZ!)**Q4n18F17$$\"3;Y:1^'H\"))GF5$\"3)H:,n??DI\"F17$$
\"3Y)fI'o`/'p\"F5$\"3A^;2tR%*)H\"F17$$\"3+Yt?JU<&y%Feq$\"3'Rw&3;\"ffH
\"F17$$!3;I(RAD\"*=Y(Feq$\"3%\\5I?ktNH\"F17$$!31=r9$>^f3#F5$\"3q6?1XIj
\"H\"F17$$!3-U?7_9](Q$F5$\"3#))[#>v!4/H\"F17$$!3jRQ/fs*pi%F5$\"3e;?qGX
()*G\"F17$$!3%fkpXsK<y&F5$\"3]\"ylDd*)**G\"F17$$!3#)\\v-dNg\"o(F5$\"3O
y%\\dLE=H\"F17$$!3Y%>;!G()=n!*F5$\"3m/3?;xP&H\"F17$$!3Z%yzWw^c`*F5$\"3
KSx=O7h(H\"F17$$!3W<$*\\AehW)*F5$\"3FMJJpi/+8F17$$!3F]TvqQcP**F5$\"3%H
&)e))=88I\"F17$$!31DKi&)H\"*))**F5$\"3a%*)*4!z)f-8F17$$!3]Jz?=#[%)***F
5$\"3(3*R%QF#*QI\"F17$$!3E?au`'Hh'**F5$\"3J%[-z!G=08F17$$!3Y$\\BUXmWQ*
F5$\"3C9g1?JI58F17$$!3b?Nz6T+@\")F5$\"3Q7UW.'>WJ\"F17$$!3'zT]Bn%\\LtF5
$\"3@4&3$HGy:8F17$$!3\"Gw;`@[CV'F5$\"3yEW0P]s;8F17$$!34_!))Qs9=V&F5$\"
39[(Rr3$><8F17$$!3+_2t&)f3ZVF5$\"3Ac;#4=TrJ\"F17$$!3QNH\\mXcmJF5$\"33N
\"G;q4lJ\"F17$$!3xGd>YliM>F5$\"3El\")H^@E:8F17$$!3trn(QF`Fr'Feq$\"3rZA
kxtP88F17$$\"3V\\fV/ivHgFeq$\"3*R\"e=vX%3J\"F17$$\"3WLop\\hp(*>F5$\"3'
*QOb+`H28F17$$\"3[A1eI[)GN$F5$\"3?E+[K/(HI\"F17$$\"3,C&zY@C<k%F5$\"3?E
:.M)**yH\"F17$$\"3ST%)=@)4(QeF5$\"33fPKD\"G@H\"F17$$\"3A^\\'pX+up(F5$
\"3Q8s%Gm*3!G\"F17$$\"3cWmtep[e!*F5$\"3SwEx;0Sm7F17$$\"3>v5$Q5&GK&*F5$
\"3=I>>#>R*e7F17$$\"3d%4lh37V%)*F5$\"3wfoT*os7D\"F17$$\"3]W&oFc'*y$**F
5$\"3iC#Q2B*QZ7F17$$\"3+$\\km`s#*)**F5$\"3Vkrlno[V7F17$$\"35F*GC!=A)**
*F5$\"31$z:vjw&R7F17$$\"3!yc)\\]jqk**F5$\"3l8%p5hpcB\"F17$$\"3)Hx+bf\"
['R*F5$\"3jm=`1&Q+A\"F17$$\"33shN?\"z9<)F5$\"3*)G=')yRO07F17$$\"3n%*4n
HO:5uF5$\"3wIiq)GC\"*>\"F17$$\"34I:;'[;*QlF5$\"3?eXT%*HG$>\"F17$$\"3w%
))>$z.pqbF5$\"3K\"\\Zhe!*y=\"F17$$\"3^IRi%fO)>XF5$\"3+'[If*3*H=\"F17$$
\"3Ef.T>nBELF5$\"3]gsn([^$y6F17$$\"3o4nH1dew?F5$\"3_6baZxMu6F17$$\"3l1
]5hpT>zFeq$\"3u@(oFO.5<\"F17$$!3i]m@d\\Zg]Feq$\"3K!)*R'44Lo6F17$$!3?c<
N]1y&y\"F5$\"3mZumNVMm6F17$$!3O(R9X)p(e.$F5$\"37`\"o5$4,l6F17$$!3jU<4'
*)*eNUF5$\"3eClkN+Jk6F17$$!3aLCQY*3]O&F5$\"3w!)H3#\\5U;\"F17$$!3]c1>'*
47ukF5$\"3[&f+M!Hsk6F17$$!3=d\"e!GeJguF5$\"3I/*GB69e;\"F17$$!3H%eLWhp[
I)F5$\"3'*y5bQ5Tn6F17$$!3'\\gc\\rZ<**)F5$\"39yjB+5Vp6F17$$!3im*4:0t7y*
F5$\"3a^p&prYP<\"F17$$!3!fiq,5uU***F5$\"3#\\H$*\\%Gby6F17$$!3@XV8mP/4&
*F5$\"3CuX**HE'R=\"F17$$!39%Hi0\"R!*y#)F5$\"3SI/,^ES)=\"F17$$!3o$)[(fa
mP](F5$\"3=W&e%)Qt)*=\"F17$$!3UX:-t\\K3mF5$\"3QGWi>-#4>\"F17$$!3*z8>sO
Npg&F5$\"3=d[ULd[\">\"F17$$!3![nEp^_c^%F5$\"3sL=]N-_\">\"F17$$!3!\\tJ9
6mXO$F5$\"3-7#Q]K\"*4>\"F17$$!37Cd13#o1;#F5$\"3YId4k6()*=\"F17$$!3#)4y
(4Uc&G#*Feq$\"3Qqb!*yz8)=\"F17$$\"3e\\4vQTU%H$Feq$\"3Wq&ezhzd=\"F17$$
\"3C\"=1Y,X(p;F5$\"3W&fqrdVD=\"F17$$\"33d)4:+[(zHF5$\"3MSgNw=fy6F17$$
\"3`v>DI(zcB%F5$\"3%y2=\")HZR<\"F17$$\"3Rh`<Z(\\ZT&F5$\"3K>#)=)>X'o6F1
7$$\"37CSIPwq*\\'F5$\"35L*>Oy2F;\"F17$$\"3)*Q1_)\\ldY(F5$\"33&y$f(G9i:
\"F17$$\"3%)ea+/6D&H)F5$\"3#fc4](RB\\6F17$$\"3odCkh0*H(*)F5$\"3Be!oh,X
=9\"F17$$\"3))yc'GQ&GZ%*F5$\"30WiUm\"R[8\"F17$$\"3!RiJA_w%y(*F5$\"3f'e
'*)oYjF6F17$$\"3m)))fP\\`()))*F5$\"3)y6ELG!)R7\"F17$$\"3%)efZHsah**F5$
\"3Lj2&[M..7\"F17$$\"3e,_M0=e'***F5$\"3N)pbF78m6\"F17$$\"3eik5BWs$***F
5$\"3rWvnO*=H6\"F17$$\"3s-uJg#)>i&*F5$\"3cs*ps])R(4\"F17$$\"3Gd))fUX<!
[)F5$\"3ki\"yATuE3\"F17$$\"3'**GVrKCmt(F5$\"3wwq$oewf2\"F17$$\"3u*>@vN
;s'oF5$\"3%*[w)30\"pp5F17$$\"37!3>r(R4')eF5$\"3Ip]VI&yQ1\"F17$$\"3qFp_
vy@4[F5$\"3')H$pl0#fe5F17$$\"3E&ywQ-*3aNF5$\"3yYWBuq]`5F17$$\"3wtzc'fO
5B#F5$\"35`$RojO\"\\5F17$$\"3A)y2QifMl)Feq$\"3Atxm'Q6b/\"F17$$!3Ke>tXj
#)o^Feq$\"3u\"*4WCxkU5F17$$!3ti)H+Kyeq\"F5$\"3\\!3Y!eiyS5F17$$!3W0LNB^
XqGF5$\"3ugL0\")*)[R5F17$$!35<Eb]E%R*RF5$\"3(>\\(4!>S(Q5F17$$!3iIO<#oe
-1&F5$\"3p6=[Ij^Q5F17$$!3kRH:sPq%>'F5$\"3***QiFKm)Q5F17$$!3>J*yU()>A@(
F5$\"3!=<6_-4)R5F17$$!3%o7JEs+O4)F5$\"3r<O(otu7/\"F17$$!3zi55\"p4A#))F
5$\"3<QCm%)Q=V5F17$$!3Fg#QCi2Ft*F5$\"39z8O32mZ5F17$$!3Cg2&[Y9k***F5$\"
3%QbK)>![F0\"F17$$!3GDH)Qr-Gg*F5$\"3gjiT<#=x0\"F17$$!3#>***4hRB$e)F5$
\"3Qy\"z^&3'>1\"F17$$!3+%o@x<it&yF5$\"3)G]Dc!ohj5F17$$!3mE%)>&)[r-qF5$
\"3WgM(zCn[1\"F17$$!3&R'odG!*HLgF5$\"3Mhb*pp\\c1\"F17$$!3!z;MC^-]'\\F5
$\"3wI%yuL4f1\"F17$$!3u%[6#[Hr\"z$F5$\"3)4*y>*p(el5F17$$!3!Q<)4J9y`DF5
$\"3E'*Q;a]jk5F17$$!3sH%e()[7BF\"F5$\"3/HIF#4BI1\"F17$$\"3/338e:r%3$Fj
[r$\"39l+gVZtg5F17$$\"3*etlsH96T\"F5$\"3guCC(pkv0\"F17$$\"3!Rr$*plcVw#
F5$\"3I6G/gij`5F17$$\"3!e1X<qdY1%F5$\"3#>J!)zhr*[5F17$$\"3uG\\3R[6(G&F
5$\"3#z)>RcqgV5F17$$\"3#*R8%*3D>]sF5$\"3:JQPDX7K5F17$$\"3-=p1cq4[()F5$
\"3A&3V$QW()=5F17$$\"3q'y%e;p_w'*F5$\"3>Psx&)Hd/5F17$$\"3C)*HSGi!*****
F5$\"3M/A4*Q0k*)*F57$$\"3!R\")*=6\\Qa'*F5$\"3WgT<o%>ot*F57$$\"3p&>&H.
\\b>')F5$\"3ms%)H8qn%e*F57$$\"3m^8!GlQn'yF5$\"3BQ='Qy!y8&*F57$$\"33Y&[
oJ%[tpF5$\"3Kcd,&4$HZ%*F57$$\"3Iw8+$RQd&fF5$\"3/c#Rp[BfQ*F57$$\"3Rew`>
)p;$[F5$\"3S*GR68!HI$*F57$$\"3#H^T\"oU\\#p$F5$\"3gATFoeb$G*F57$$\"3;5G
>0>r%\\#F5$\"3YV$[<p@FC*F57$$\"3'=LgRyLtD\"F5$\"3#*eWFKb.3#*F57$$!3e=8
odE<Le!#D$\"3]'*****Rxkz\"*F5-%&COLORG6&%$RGBG\"\"!$\"\"(!\"\"Fjgx-%*T
HICKNESSG6#\"\"#-F(6%7$7$$FjgxFjgx$\"37+++++SrIF17$Ffhx$\"3;+++++++KF1
FfgxF^hx-F(6%7$7$$!\"$FjgxFjhx7$$\"\"$FjgxFjhx-%'COLOURG6&Figx$\")#)eq
k!\")$\"))eqk\"FjixF[jxF^hx-F(6%7$7$Ffhx$\"3%)******Rxkz\"*F57$Ffhx$\"
31+++Sxkz\")F5FfgxF^hx-%)POLYGONSG6$7&7$$F]hxFjgx$\"*uZ'z\")!\"*7$$\"
\"\"FjgxF\\[y7$F`[y$\"*uZ'zJF^[y7$F[[yFc[y-Fggx6&Figx$\"\"'F]hxFh[y$\"
\")F]hx-F(6&7#7$Ffhx$\"31+++SxkzcF5-%'SYMBOLG6#%'CIRCLEG-Ffix6&FigxFfh
xFfhx$\"*++++\"Fjix-%&STYLEG6#%&POINTG-F(6&F^\\y-Fc\\y6#%(DIAMONDGFf\\
yFj\\y-F(6&F^\\y-Fc\\y6#%&CROSSGFf\\yFj\\y-F(6$7_xF_\\y7$$\"3$*****\\i
lyM;Feq$\"3;r^Yw#30^'F57$$\"3')*****\\7t&pKFeq$\"3_8Co[K@>tF57$$\"3z**
**\\(ofV!\\Feq$\"31y$Q$))y>%3)F57$$\"3s******\\i9RlFeq$\"3O\\-%\\Ej]y)
F57$$\"3g+](oHl:'zFeq$\"3)3)R&G[;\"G$*F57$$\"3Y,+vVV)RQ*Feq$\"33*eS#*[
vuz*F57$$\"3A+D1R.k!3\"F5$\"3sD7q&)fO=5F17$$\"3/++vVA)GA\"F5$\"3Yq`g<p
)y/\"F17$$\"3C+v=U\"[GQ\"F5$\"3oXVj_L[p5F17$$\"3;+]iSS\"Ga\"F5$\"31!p9
sst#y5F17$$\"3)*\\P%)*)pzA;F5$\"3UjE^5Nyx5F17$$\"3!)*\\i!R*zFq\"F5$\"3
gWa#HeLS2\"F17$$\"3!*\\7G))Gw#y\"F5$\"37$Hu4#z/n5F17$$\"3+++]Peui=F5$
\"3Ix4-!>ro0\"F17$$\"31++v$4sP-#F5$\"3PzA/N\"))p-\"F17$$\"3S+++]$)z%=#
F5$\"3A(4TN]&G_)*F57$$\"3W++D1Y#eM#F5$\"3^m$*\\:!HnK*F57$$\"3A++]i3&o]
#F5$\"3\">&=Cf)4oq)F57$$\"37+D1*y6rm#F5$\"3\\7ZbTO/7!)F57$$\"3.+]i:FPF
GF5$\"3!fA_iG,vD(F57$$\"3[+v=UOj()HF5$\"3eL@g)o?DY'F57$$\"3S++voX*y9$F
5$\"3[SY2@hZZcF57$$\"35++]ipZ'H$F5$\"3[*zhbF2@*[F57$$\"3#)***\\iNf]W$F
5$\"3c>&3Ry#4aTF57$$\"31+++]<k$f$F5$\"3C'4$eQcp\\MF57$$\"3z***\\P9CAu$
F5$\"3#)[8UN!QWz#F57$$\"33++D1E2'*QF5$\"3YB2a)oAJ=#F57$$\"3!)***\\(o5#
*\\SF5$\"3GL*GIq/Wl\"F57$$\"33++DJ&pP?%F5$\"3L=l4DQx?7F57$$\"3O++v$*zh
dVF5$\"3Sa-knetC*)Feq7$$\"3H++v$HGn^%F5$\"3]JC*3Kw'>nFeq7$$\"3A++v$fQe
n%F5$\"3KLv%[<$pzdFeq7$$\"3Y++vVPRbZF5$\"3_utCrkz#z&Feq7$$\"39++v$*)[
\\$[F5$\"3UZ\"=U'H`GhFeq7$$\"3Q++vVS]9\\F5$\"3!Rc71*)yZy'Feq7$$\"31++v
$>fS*\\F5$\"3E\"R6d/$QdxFeq7$$\"3S,++D#fE:&F5$\"3G>olE')Rh5F57$$\"3^++
Dc#f7J&F5$\"3]1$Qj%H)HY\"F57$$\"3s++](Gf)paF5$\"3KN8D!)4Tq>F57$$\"3$)*
**\\(=$f%GcF5$\"3/%)obJc%4d#F57$$\"3_+DJX*)f\"z&F5$\"3!yrNOd=*pKF57$$
\"35+](=dQZ&fF5$\"3c>gqGK)G.%F57$$\"3!3]P%)>yy6'F5$\"3a*=T8ky&R[F57$$
\"3Q+++Dy,\"G'F5$\"3Q%*yDhDeocF57$$\"3w**\\(oH6ZU'F5$\"3U-6.))3u*R'F57
$$\"3C++voZSolF5$\"3k%>j6Vcg6(F57$$\"3r+]iS#)47nF5$\"3B<'y@FkF!yF57$$
\"3=,+]7<zboF5$\"3'H`adx4dW)F57$$\"3v+]7Glb<qF5$\"3ioJ#>Hs15*F57$$\"3U
,+vV8KzrF5$\"3kGnhv*3jm*F57$$\"3)4+v$fh3TtF5$\"3F\\0md]y75F17$$\"3`+++
v4&G](F5$\"3yEX\\&eCt/\"F17$$\"3!4++D,!GlwF5$\"3i\\k[V8Tp5F17$$\"3<+++
]!4x#yF5$\"3'REA;S(Hy5F17$$\"3N++voN#*3zF5$\"33J'G#fKqx5F17$$\"3U****
\\(3Q,*zF5$\"3)4G9B$)[P2\"F17$$\"3i***\\ig_82)F5$\"3-@u9*=gk1\"F17$$\"
3!)*****\\7nD:)F5$\"3O$>GxP&)e0\"F17$$\"3U*******f(44$)F5$\"3IE9V>*3l-
\"F17$$\"39+++v!GcY)F5$\"3U:\"HMI-#f)*F57$$\"3'3+++be@i)F5$\"3W+-kqx6^
$*F57$$\"3[+++D!*oy()F5$\"3w$*p\\y\"fKv)F57$$\"3*3](=#pL3#*)F5$\"3v6'H
=64[9)F57$$\"3=+]Pf$yH1*F5$\"3shC&p+Mm[(F57$$\"3f+DcEI70#*F5$\"3e.s]&)
*4?z'F57$$\"3+,+v$pnsM*F5$\"3'*zn>%Q[\\2'F57$$\"3y+Dc^9H;&*F5$\"3Ydi3F
^68_F57$$\"3e+]P4_J&o*F5$\"3u$)f8`&yXO%F57$$\"3O+v=n*QV&)*F5$\"3gZl#3(
G_`NF57$$\"3,++]siL-5F1$\"3-30wUS1.GF57$$\"3!***\\(=![l;5F1$\"3&eC!=T4
oIAF57$$\"3,++DJL(4.\"F1$\"3g?8hFz))G<F57$$\"37+]ig=HX5F1$\"3jSA'>@bzI
\"F57$$\"3-+++!R5'f5F1$\"3ir%>WM!)\\w*Feq7$$\"3+]PfViEw5F1$\"3O'fmZg]S
7(Feq7$$\"3++v=(4AH4\"F1$\"3/k\\\"[(z$z&eFeq7$$\"3*\\P%)R-]75\"F1$\"3c
5k?/$4Hv&Feq7$$\"3x\\7y]zd46F1$\"353R(R]&o,gFeq7$$\"3*\\7yv(e!z6\"F1$
\"35<6a*>UDg'Feq7$$\"3)***\\P/QBE6F1$\"3?huN&f98b(Feq7$$\"3%**\\(o4.sb
6F1$\"35ufzZ%e(e8F57$$\"3!******\\\"o?&=\"F1$\"3p9OVxFNNBF57$$\"37]P%)
*\\#Q,7F1$\"3c\\/V,I@+IF57$$\"35+vo%=ev@\"F1$\"3&*\\&RpkG]t$F57$$\"34]
7`pQtL7F1$\"3b_WKW\\h?XF57$$\"31+]Pa&4*\\7F1$\"3()p*f(*ehkL&F57$$\"33+
DcJEJl7F1$\"3]Y\"Q['y)>7'F57$$\"33++v3dr!G\"F1$\"3EFtoT,/(*oF57$$\"31+
v$fy=hH\"F1$\"3\"Q!pfhiEVwF57$$\"33+]7j=_68F1$\"3!*HU6Hm*HM)F57$$\"3.]
7GeLfF8F1$\"3$3G@wDCc+*F57$$\"3(**\\PM&[mV8F1$\"3[&)GJ2!HDe*F57$$\"39]
Pf[jtf8F1$\"3^Kj3)=%)e+\"F17$$\"33++vVy!eP\"F1$\"3e&)e0^'GA/\"F17$$\"3
E]7G$\\m0R\"F1$\"39rA2sY&[1\"F17$$\"3@+D\"G9D`S\"F1$\"3Og<<\"=xm2\"F17
$$\"3HD\"yvY/FT\"F1$\"3[L<HFuWy5F17$$\"3:]PM#z$3?9F1$\"3M5nAI\"Ru2\"F1
7$$\"3Bv$4r6juU\"F1$\"3#o;5\"yxlt5F17$$\"3K+](=WU[V\"F1$\"3%RtQF&R7n5F
17$$\"37](=#R;w]9F1$\"3`\\s?uI%Q/\"F17$$\"3Q+DcO3om9F1$\"3r7IQ%*y_35F1
7$$\"3=]i!R.+E[\"F1$\"3E^f;^[r?'*F57$$\"3?++DJ#>&)\\\"F1$\"3q#fxZk'[c!
*F57$$\"33]7G.[0::F1$\"3O9aw)46-Q)F57$$\"3(**\\7`P!fJ:F1$\"3PvEVbMEIwF
57$$\"33]PMZf7[:F1$\"3,<N!*)[-r#oF57$$\"3)***\\P>:mk:F1$\"3E.97Z'QE*fF
57$$\"33+vV$yb!z:F1$\"3+34Z]WZe_F57$$\"3=++]Z+X$f\"F1$\"3@C')***o@I`%F
57$$\"32+Dc6V%yg\"F1$\"39#pTuV&GJQF57$$\"3'***\\iv&QAi\"F1$\"3%HM-'=-y
nJF57$$\"3)*\\i:vZyP;F1$\"3i&)*\\l%=#*4DF57$$\"3-+vou4L`;F1$\"3P'eWNh;
&G>F57$$\"3!)\\(=U<x)o;F1$\"3i#\\rY*yeP9F57$$\"3G++vtLU%o\"F1$\"3MgX.!
zw*[5F57$$\"3>+D1>T[+<F1$\"3ab*GGRe)[wFeq7$$\"34+]Pk[a;<F1$\"3!=!eTfo(
H2'Feq7$$\"3P]7.P_dC<F1$\"3)Q0!)>g.Mx&Feq7$$\"3++vo4cgK<F1$\"3\"oX%3ta
o-eFeq7$$\"3$)\\PM#)fjS<F1$\"3E_q^WPjghFeq7$$\"37+++bjm[<F1$\"3>Zqcf8%
\\%oFeq7$$\"3C+](o'44!y\"F1$\"3![CTKb=[D\"F57$$\"38++vyb^6=F1$\"3[>y=I
O^eAF57$$\"3:]ilU!=n#=F1$\"3a)*ou38!=(GF57$$\"3<+Dc10#>%=F1$\"35rOe=w&
)\\NF57$$\"3=](o/(H7d=F1$\"37jwqD=/xUF57$$\"3)***\\PMaKs=F1$\"3ciHEi*z
l.&F57$$\"3-]Pfy^?*)=F1$\"3ADdBzv_'*eF57$$\"33+D\"G#\\31>F1$\"3MK'z#*Q
5.v'F57$$\"3\"*\\7.nY'H#>F1$\"3G6k'eifOd(F57$$\"3&****\\7TW)R>F1$\"3S^
BW'frJM)F57$$\"33+v=7;,b>F1$\"37\\\"35gA1(*)F57$$\"3)***\\78)y,(>F1$\"
3vvj/I<^A&*F57$$\"3m*\\iS,Y`)>F1$\"3qBy!fUnh)**F57$$\"3z*****\\@80+#F1
$\"3%4c')**R%4N5F17$$\"3>+]PRuq;?F1$\"3'zo8AW'=i5F17$$\"3q***\\Pm,H.#F
1$\"3gfhpycMw5F17$$\"37+v$fx)*4/#F1$\"3e$zX1;Y%y5F17$$\"35+]7))e4\\?F1
$\"3]5v:!f,s2\"F17$$\"3k*\\7.+$>d?F1$\"3k$ozD7?E2\"F17$$\"31++]7,Hl?F1
$\"3GzQ3z<tk5F17$$\"3=+v$4')QY4#F1$\"3CUN+_[X45F17$$\"3()**\\P4w)R7#F1
$\"3C+m^lRAk\"*F57$$\"3;]iSw0.S@F1$\"3vzhhuR\\B&)F57$$\"3-+vVVN2c@F1$
\"3Gg#HU$os4yF57$$\"3K](o/^;@<#F1$\"3caz9')QDTqF57$$\"3g++]x%f\")=#F1$
\"3#fX*Qb9\"yB'F57$$\"33](=#fYD.AF1$\"3s;exxvOoaF57$$\"3V+v$4%)\\$=AF1
$\"3$R_3/*)GPq%F57$$\"3M]ilA]WLAF1$\"3)=X$oNdGhRF57$$\"3C+]P/-a[AF1$\"
3+@f@eO#zD$F57$$\"3?++v=!>VE#F1$\"3WM?)e,F=e#F57$$\"39+]7Ly4!G#F1$\"37
O%[SB(p#)>F57$$\"33++]Zm(eH#F1$\"3GS<9$)*>aZ\"F57$$\"3/+](=Yb;J#F1$\"3
%f'\\%G\"))fs5F57$$\"3A]7.P@3FBF1$\"3O&=$R\"*fr$*yFeq7$$\"3&**\\(=7)3D
M#F1$\"3F\"*=9\\o$HA'Feq7$$\"3EDcw\\@A]BF1$\"3s*GG7+;m$eFeq7$$\"39]PM(
[NzN#F1$\"3A@b(p\\LLv&Feq7$$\"3M7G8c@zhBF1$\"3$37-a$pbDeFeq7$$\"3cu=#
\\#)[cO#F1$\"3Ws\"H[e%etfFeq7$$\"3AP4r$\\0&pBF1$\"3ELpy(H'>(>'Feq7$$\"
3')****\\i@OtBF1$\"33lO!y^fg\\'Feq7$$\"34+v$4wicS#F1$\"3oNdn[2c'=\"F57
$$\"3I+]PfL'zV#F1$\"3/u69-1B)=#F57$$\"31]7.<!=NX#F1$\"3i1Ab9IQ2GF57$$
\"3E+vouE2pCF1$\"3GK!)y:\"*)e\\$F57$$\"3W]PMKti%[#F1$\"3nw.=TU7PUF57$$
\"3>+++!*>=+DF1$\"3m#QB8`!>8]F57$$\"3[+D1u))3;DF1$\"3IxAF$HCL#eF57$$\"
3L+]7ed*>`#F1$\"37TqW\"GI)HmF57$$\"3=+v=UE!za#F1$\"3&\\xL>bWBT(F57$$\"
3[++DE&4Qc#F1$\"3!GS0=%*36:)F57$$\"3G]7G$p%ezDF1$\"3'*)4S/Rt@#))F57$$
\"3a+DJg)f`f#F1$\"3)H&4))zi>:%*F57$$\"3O]PMF]86EF1$\"3%3*\\7l/X:**F57$
$\"3=+]P%>5pi#F1$\"3D'zW@E^5.\"F17$$\"3R]7`MfSTEF1$\"3_@(*3;GBd5F17$$
\"3g+vou;!fl#F1$\"3Cj&>Ca^J2\"F17$$\"3EDcwW&\\Jm#F1$\"3GcjNm$\\r2\"F17
$$\"3O]P%[T(RqEF1$\"3w&4X=`t%y5F17$$\"3[v=#\\GXwn#F1$\"3))RW(e3<r2\"F1
7$$\"3e+++bJ*[o#F1$\"318h&4:(3t5F17$$\"3d+D14p],FF1$\"3Q`\\Dk.!R0\"F17
$$\"3c+]7j17=FF1$\"3qxPIg<L@5F17$$\"35+v=<WtMFF1$\"311lf:Ayi(*F57$$\"3
a++Dr\"[8v#F1$\"3+YEtPW!)*>*F57$$\"3_+v$4r2iw#F1$\"3U1$\\HIAOh)F57$$\"
3/+]i]s1\"y#F1$\"3=**Q%3RvF'zF57$$\"3-+DJ!zEfz#F1$\"3u)))oCO3;E(F57$$
\"3++++Ijy5GF1$\"37?7'Q$\\dDlF57$$\"3Y]Pf)pHm#GF1$\"35%H`?Ow1s&F57$$\"
3.+v=nIZUGF1$\"3_=ppt+v9\\F57$$\"31]7yNkJeGF1$\"3z'oR;`$)z7%F57$$\"3_+
]P/)fT(GF1$\"3!\\IO1F%3!Q$F57$$\"3S+voHRK*)GF1$\"3G\"z_)o<'zr#F57$$\"3
H+++b!)[/HF1$\"3CubAPI\"Q7#F57$$\"3=+DJ!=_'>HF1$\"38N8X1WF6;F57$$\"3^+
]i0j\"[$HF1$\"3s-8iq\"4@>\"F57$$\"3Q](=#HA6^HF1$\"3!4L_)[?(oc)Feq7$$\"
3E+D\"G:3u'HF1$\"3]^,!*>@f!\\'Feq7$$\"3%y$4rL@[rHF1$\"39!e7p,;%yhFeq7$
$\"3Uv$4Y6cb(HF1$\"3;3!3GrS-&fFeq7$$\"3c7y]&4I'zHF1$\"3%\\ir]'[W1eFeq7
$$\"39]iSwSq$)HF1$\"3wO2F.rEZdFeq7$$\"3s(o/t0yx)HF1$\"3m0%[Ij0Gx&Feq7$
$\"3GDJ?Q?&=*HF1$\"3op3:t!=I)eFeq7$$\"3Ui:5>g#f*HF1$\"33$Qfd_@x2'Feq7$
Fcix$\"3G['4+)GfcjFeq-Ffix6&FigxFh\\yFfhxFfhx-%+AXESLABELSG6%Q!6\"Fh_]
l-%%FONTG6#%(DEFAULTG-%%VIEWG6$F]`]lF]`]l7-F'FbhxF\\ixF]jxFfjxF\\\\yF^
]yFc]y-F(6$7_x7$Ffhx$\"3!fr6etZ'zcF57$F\\^y$\"3xEFLs#30^'F57$Fa^y$\"3l
4grWK@>tF57$Ff^y$\"3)yxVY)y>%3)F57$F[_y$\"3sqfhhK1&y)F57$F`_y$\"3q[W#*
zk6G$*F57$Fe_y$\"39x\\w'[vuz*F57$Fj_y$\"3LcS]&)fO=5F17$F_`y$\"3+wDY<p)
y/\"F17$Fd`y$\"31!4cDN$[p5F17$Fi`y$\"3)=(H?FPFy5F17$F^ay$\"35LW`5Nyx5F
17$Fcay$\"3si0)HeLS2\"F17$Fhay$\"3a2C1@z/n5F17$F]by$\"3UC:9!>ro0\"F17$
Fbby$\"3vwbAN\"))p-\"F17$Fgby$\"35[T&f]&G_)*F57$F\\cy$\"3(*f+V=!HnK*F5
7$Facy$\"3'>O9E')4oq)F57$Ffcy$\"3w;.GXO/7!)F57$F[dy$\"3+VaB!H,vD(F57$F
`dy$\"3Ox3u#p?DY'F57$Fedy$\"3QTGEDhZZcF57$Fjdy$\"3iZ**pzs5#*[F57$F_ey$
\"3qDa!zy#4aTF57$Fdey$\"3g$o]Bk&p\\MF57$Fiey$\"3!\\fw)Q!QWz#F57$F^fy$
\"3-CCf\"pAJ=#F57$Fcfy$\"3?)*\\g0ZSa;F57$Fhfy$\"3<&4Or#Qx?7F57$F]gy$\"
3XF$*=#)etC*)Feq7$Fbgy$\"3^i].Hjn>nFeq7$Fggy$\"3EJ!yj<$pzdFeq7$F\\hy$
\"3SqnWpkz#z&Feq7$Fahy$\"38Ky4fH`GhFeq7$Ffhy$\"3mA\\?#))yZy'Feq7$F[iy$
\"3%[hpS.$QdxFeq7$F`iy$\"3WS=([i)Rh5F57$Feiy$\"3kIt(R%H)HY\"F57$Fjiy$
\"3eDOPx4Tq>F57$F_jy$\"3KuYBGc%4d#F57$Fdjy$\"3e)QV*p&=*pKF57$Fijy$\"3'
=iTZA$)G.%F57$F^[z$\"3E++@P'y&R[F57$Fc[z$\"3A?'pqb#eocF57$Fh[z$\"3k$p%
)Q)3u*R'F57$F]\\z$\"3N<T9Fk0;rF57$Fb\\z$\"3/9(p$oUw-yF57$Fg\\z$\"3_:VB
s(4dW)F57$F\\]z$\"3Q0W\"))Gs15*F57$Fa]z$\"3M\"f+I(*3jm*F57$Ff]z$\"3BH]
Xd]y75F17$F[^z$\"3om/N&eCt/\"F17$F`^z$\"3C%)ySV8Tp5F17$Fe^z$\"3Uh7h,uH
y5F17$Fj^z$\"3oI;DfKqx5F17$F__z$\"3GJ6PK)[P2\"F17$Fd_z$\"3+ZxB*=gk1\"F
17$Fi_z$\"3Y*R^yP&)e0\"F17$F^`z$\"3EHbh>*3l-\"F17$Fc`z$\"33'4MeI-#f)*F
57$Fh`z$\"3#fK]Nx<6N*F57$F]az$\"3_16%==fKv)F57$Fbaz$\"32Hs\\:\"4[9)F57
$Fgaz$\"3o,'p3,Mm[(F57$F\\bz$\"3!\\&[f*)*4?z'F57$Fabz$\"3i)[s$)Q[\\2'F
57$Ffbz$\"3k:qDJ^68_F57$F[cz$\"3')>H=d&yXO%F57$F`cz$\"3g?VjuG_`NF57$Fe
cz$\"3N\"f?i/kI!GF57$Fjcz$\"3+e\"oU%4oIAF57$F_dz$\"37iQEIz))G<F57$Fddz
$\"3]?^79_&zI\"F57$Fidz$\"3!e)*32O!)\\w*Feq7$F^ez$\"3ITOV910CrFeq7$Fce
z$\"3?)*>fxz$z&eFeq7$Fhez$\"3;Lv\\.$4Hv&Feq7$F]fz$\"3`[Sy*\\&o,gFeq7$F
bfz$\"3p2$**=>UDg'Feq7$Fgfz$\"3M(o;Ve98b(Feq7$F\\gz$\"3MQacX%e(e8F57$F
agz$\"3)[:pUx_`L#F57$Ffgz$\"3^=a'y*H@+IF57$F[hz$\"3UDq1V'G]t$F57$F`hz$
\"37QbCS\\h?XF57$Fehz$\"3Ng6e&ehkL&F57$Fjhz$\"3=9cmgy)>7'F57$F_iz$\"3+
s)>w8Sq*oF57$Fdiz$\"3XP3tdiEVwF57$Fiiz$\"3y%4Tbi'*HM)F57$F^jz$\"3i`QWa
Ui0!*F57$Fcjz$\"3WeIh/!HDe*F57$Fhjz$\"3[o5(y=%)e+\"F17$F][[l$\"3B))3!4
lGA/\"F17$Fb[[l$\"3Mvh(>na[1\"F17$Fg[[l$\"37bm8\"=xm2\"F17$F\\\\[l$\"3
)R\\(GFuWy5F17$Fa\\[l$\"3ofLDI\"Ru2\"F17$Ff\\[l$\"3efv;yxlt5F17$F[][l$
\"3#zbEG&R7n5F17$F`][l$\"3!)z)eV2VQ/\"F17$Fe][l$\"3y9Yf%*y_35F17$Fj][l
$\"3f!=GQ&[r?'*F57$F_^[l$\"3]/(=zk'[c!*F57$Fd^[l$\"3!zp>B56-Q)F57$Fi^[
l$\"3SqJIfMEIwF57$F^_[l$\"3oMY)H\\-r#oF57$Fc_[l$\"3G4=I^'QE*fF57$Fh_[l
$\"3q9\\kaWZe_F57$F]`[l$\"3'y*)zSp@I`%F57$Fb`[l$\"3%p\"eMTaGJQF57$Fg`[
l$\"3!oc[AA!ynJF57$F\\a[l$\"3OaJ$)\\=#*4DF57$Faa[l$\"3U4bQ;m^G>F57$Ffa
[l$\"3p[;+(*yeP9F57$F[b[l$\"31Yxz\"zw*[5F57$F`b[l$\"3]4&eTSe)[wFeq7$Fe
b[l$\"3!fV^T'o(H2'Feq7$Fjb[l$\"39PAO.OStdFeq7$F_c[l$\"3B&G/6Z&o-eFeq7$
Fdc[l$\"3I#G(=RPjghFeq7$Fic[l$\"3;H?#4NT\\%oFeq7$F^d[l$\"3kfB:^&=[D\"F
57$Fcd[l$\"3GP\"zqi8&eAF57$Fhd[l$\"3m<!\\_I,=(GF57$F]e[l$\"3-xsx9w&)\\
NF57$Fbe[l$\"3!yb!o@=/xUF57$Fge[l$\"3gS!3\"e*zl.&F57$F\\f[l$\"3[A70vv_
'*eF57$Faf[l$\"3c#\\%=&Q5.v'F57$Fff[l$\"3CSq(>ifOd(F57$F[g[l$\"350$pGf
rJM)F57$F`g[l$\"3!fG1yfA1(*)F57$Feg[l$\"3.4#*GF<^A&*F57$Fjg[l$\"3ig'eO
Unh)**F57$F_h[l$\"3?4w\")*R%4N5F17$Fdh[l$\"3eQ(3@W'=i5F17$Fih[l$\"3/fz
lycMw5F17$F^i[l$\"3/`9kghWy5F17$Fci[l$\"3=eq=!f,s2\"F17$Fhi[l$\"3sDHkA
,is5F17$F]j[l$\"33-/=z<tk5F17$Fbj[l$\"3,SQ@_[X45F17$Fgj[l$\"3M([x&oRAk
\"*F57$F\\[\\l$\"3mHW4yR\\B&)F57$Fa[\\l$\"3YWb.Qos4yF57$Ff[\\l$\"3(*=X
=!*QDTqF57$F[\\\\l$\"3GJEbf9\"yB'F57$F`\\\\l$\"3k70'>en$oaF57$Fe\\\\l$
\"3bm&>X*)GPq%F57$Fj\\\\l$\"3wVtiRdGhRF57$F_]\\l$\"3!R'H!>mBzD$F57$Fd]
\\l$\"3gx5@>q#=e#F57$Fi]\\l$\"3!zwOpB(p#)>F57$F^^\\l$\"3o)f<b)*>aZ\"F5
7$Fc^\\l$\"3%4K\\Y\"))fs5F57$Fh^\\l$\"3)zv9M+;P*yFeq7$F]_\\l$\"3$\\'f&
[&o$HA'Feq7$Fb_\\l$\"3,!>GP+;m$eFeq7$Fg_\\l$\"3C#QYi\\LLv&Feq7$F\\`\\l
$\"36%))eI$pbDeFeq7$Fa`\\l$\"33C`(3e%etfFeq7$Ff`\\l$\"3Wd$GAH'>(>'Feq7
$F[a\\l$\"3'=i[1^fg\\'Feq7$F`a\\l$\"3MPtoY2c'=\"F57$Fea\\l$\"3URb3*fI#
)=#F57$Fja\\l$\"364(*36IQ2GF57$F_b\\l$\"39wA+7\"*)e\\$F57$Fdb\\l$\"3vt
F;PU7PUF57$Fib\\l$\"30U4<F0>8]F57$F^c\\l$\"3\"=l&3*GCL#eF57$Fcc\\l$\"3
'G%>Lx-$)HmF57$Fhc\\l$\"3Z2T*zaWBT(F57$F]d\\l$\"3'4kS\"Q*36:)F57$Fbd\\
l$\"3:y$RrQt@#))F57$Fgd\\l$\"3=Vi-xi>:%*F57$F\\e\\l$\"3M&=(yi/X:**F57$
Fae\\l$\"3;7&o>E^5.\"F17$Ffe\\l$\"31m,(f\"GBd5F17$F[f\\l$\"39W#fBa^J2
\"F17$F`f\\l$\"314iKm$\\r2\"F17$Fef\\l$\"3sy_%=`t%y5F17$Fjf\\l$\"3#>&
\\!f3<r2\"F17$F_g\\l$\"3'Qz;5:(3t5F17$Fdg\\l$\"37JLQk.!R0\"F17$Fig\\l$
\"3p<j\\g<L@5F17$F^h\\l$\"3i+06=Ayi(*F57$Fch\\l$\"3A1gwSW!)*>*F57$Fhh
\\l$\"3Q,oP1Bi8')F57$F]i\\l$\"3G*)**e%RvF'zF57$Fbi\\l$\"3%*35Xm$3;E(F5
7$Fgi\\l$\"3/0;*z$\\dDlF57$F\\j\\l$\"3#GWTiOw1s&F57$Faj\\l$\"3%Q#z$y2]
Z\"\\F57$Ffj\\l$\"3BA)Hc`$)z7%F57$F[[]l$\"3a@cPuU3!Q$F57$F`[]l$\"3V2UE
s<'zr#F57$Fe[]l$\"3w%zI-/8Q7#F57$Fj[]l$\"3u&Q\")*3WF6;F57$F_\\]l$\"3a;
\"=E<4@>\"F57$Fd\\]l$\"3QvGei?(oc)Feq7$Fi\\]l$\"3-r#Hq7#f!\\'Feq7$F^]]
l$\"35g[NAgTyhFeq7$Fc]]l$\"3C+Wb;2C]fFeq7$Fh]]l$\"3!>\"f6n[W1eFeq7$F]^
]l$\"3_V&4O5nsu&Feq7$Fb^]l$\"3#\\:\"oJc!Gx&Feq7$Fg^]l$\"3OP)z+2=I)eFeq
7$F\\_]l$\"37l'*)4_@x2'Feq7$Fcix$\"3eg\"\\N(GfcjFeqFc_]lFe_]lF^`]l-%*A
XESSTYLEG6#%%NONEG-F_`]l6$;F`ixFcix;$!\"&F]hx$\"#NF]hx" 1 2 0 1 10 0 
2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+
3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" }}}
{PARA 0 "" 0 "" {TEXT -1 79 "The following code will generate an anima
tion to illustrate oscillatory motion." }}{PARA 0 "" 0 "" {TEXT -1 
121 "It requires a fairly large amount of memory, so you should probab
ly not save this worksheet with the animation displayed." }}{PARA 0 "
" 0 "" {TEXT -1 215 "The values of the variables introduced in the fir
st few lines can be adjusted to alter the appearance of the picture, a
nd the amount of memory required. (The picture above is just the first
 frame of the animation.) " }}{PARA 0 "" 0 "" {TEXT -1 156 "Once gener
ated, the animation can be played by clicking on the picture and using
 the animation menu which appears, or the controls in the context bar.
 See: " }{HYPERLNK 17 "Context Bar for Animations" 2 "worksheet/refere
nce/contextanimate" "" }{TEXT -1 4 ".   " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1259 "numframes := 30:\ns
pring_points := 150:\nspring_colour := COLOR(RGB,0,.7,0):\nweight_colo
ur := COLOR(RGB,.6,.6,.8):\nsupport_colour := brown:\npoint_colour := \+
blue:\ngraph_colour := red:\nfrms := NULL:\nphi := evalf(2*Pi/numframe
s):\nfor i from 0 to numframes-1 do   \n   a := evalf(-Pi/2);\n   b :=
 evalf(32.5*Pi);\n   c := 2.61-0.02*b;\n   d := (0.02+0.005*sin(i*phi)
);\n   e := 0.005*b;\n   db := d*b;\n   p1 := plot([cos(t),3-0.04*sin(
t)-d*t,t=a..b],\n             color=spring_colour,thickness=2,numpoint
s=spring_points);\n   p2 := plot([[0,3.0714],[0,3.2]],color=spring_col
our,thickness=2);\n   p3 := plot([[-3,3.2],[3,3.2]],color=support_colo
ur,thickness=2);\n   p4 := plot([[0,2.96-db],[0,2.86-db]],color=spring
_colour,thickness=2);\n   p5 := plots[polygonplot]([[-1,2.86-db],[1,2.
86-db],\n    [1,2.36-db],[-1,2.36-db]],color=weight_colour);\n   p6 :=
 plot([[[0,2.61-db]]$3],style=point,\n              symbol=[circle,dia
mond,cross],color=point_colour);\n   p7 := plot([t/10,c+e*sin(t-i*phi)
,t=0..30],color=graph_colour);\n   frms := frms,plots[display]([p1,p2,
p3,p4,p5,p6,p7]);\nend do:\nfrms:= [frms]:\nunassign('a','b','c','d','
e','db','phi',\n                      'p1','p2','p3','p4','p5','p6','p
7'):\nplots[display](frms,insequence=true,view=[-3..3,-0.5..3.5],\n   \+
 axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 22 "Simple harmonic motion" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{GLPLOT2D 406 99 99 {PLOTDATA 2 "6'-%'CURVESG6%7D7$\"\"!$\"\"&!\"\"7$$
F*!\"#F)7$$\"\"\"F+$\"\")F+7$$\"\"#F+F57$$\"\"$F+F27$$\"\"%F+F57$F)F27
$$\"\"'F+F57$$\"\"(F+F27$F2F57$$\"\"*F+F27$F1F57$$\"#6F+F27$$\"#7F+F57
$$\"#8F+F27$$\"#9F+F57$$\"#:F+F27$$\"#;F+F57$$\"#<F+F27$$\"#=F+F57$$\"
#>F+F27$F6F57$$\"#@F+F27$$\"#AF+F57$$\"#BF+F27$$\"#CF+F57$$\"#DF+F27$$
\"#EF+F57$$\"#FF+F27$$\"#GF+F57$$\"#HF+F27$F9F57$$\"$0$F.F)7$$\"#JF+F)
-%*THICKNESSG6#F6-%&COLORG6&%$RGBGF(F?F(-F$6%7%7$F(F17$F(F(7$F<F(Faq-F
eq6&FgqF5F5F5-%)POLYGONSG6$7'7$F_q$F6F.7$F_q$\"#)*F.7$$\"#OF+Fgr7$FjrF
erFdr-Feq6&FgqF2F2F2-%%TEXTG6$7$$\"$N$F.F)%\"MG-%*AXESSTYLEG6#%%NONEG
" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 43.000000 0 0 "Curve 1" "C
urve 2" "Curve 3" "Curve 4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 153 "Suppose, as in the picture above, that a mass \+
M is attached to a spring, which, in turn, is connected to a rigid sup
port at the opposite end to the mass." }}{PARA 0 "" 0 "" {TEXT -1 38 "
Suppose also that the mass rests on a " }{TEXT 259 20 "frictionless su
rface" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 3 "By " }{TEXT 259 11 "Hooke's law" }{TEXT -1 214 ", if th
e mass is displaced to the right or left from its equilibrium position
, the spring exerts a restoring force in the opposite direction to the
 displacement, which is proportional to the amount of displacement." }
}{PARA 0 "" 0 "" {TEXT -1 21 "Thus if the force is " }{TEXT 267 1 "F" 
}{TEXT -1 25 " and the displacement is " }{TEXT 266 1 "x" }{TEXT -1 5 
" then" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "F = -k x" "6#/%\"FG,$*&%\"k
G\"\"\"%\"xGF(!\"\"" }{TEXT -1 13 "  ------- (i)" }}{PARA 0 "" 0 "" 
{TEXT -1 6 "where " }{TEXT 269 1 "k" }{TEXT -1 111 " is a constant. Th
e minus sign is reuired because the force acts in the opposite directi
on to the displacement." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 29 "If the mass is set in motion " }{TEXT 259 29 "Newton
's second law of motion" }{TEXT -1 60 " says that \"force equals mass \+
times acceleration\", that is, " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "F \+
= M;" "6#/%\"FG%\"MG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2);" 
"6#*(%\"dG\"\"#%\"xG\"\"\"*&F$F'*$%\"tGF%F'!\"\"" }{TEXT -1 14 "  ----
--- (ii)" }}{PARA 0 "" 0 "" {TEXT -1 7 "where  " }{XPPEDIT 18 0 "d^2*x
/(d*t^2);" "6#*(%\"dG\"\"#%\"xG\"\"\"*&F$F'*$%\"tGF%F'!\"\"" }{TEXT 
-1 9 "  is the " }{TEXT 259 12 "acceleration" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 71 "Equations (i) and (ii) give rise to the followi
ng differential equation" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "d^2*x/(d*t^2)+k/M;" "6#,&*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*
$%\"tGF&F(!\"\"F(*&%\"kGF(%\"MGF,F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x
 = 0;" "6#/%\"xG\"\"!" }{TEXT -1 15 "  ------- (iii)" }}{PARA 0 "" 0 "
" {TEXT -1 37 "which governs the motion of the mass " }{TEXT 268 1 "M
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 41 "Equation (iii) can be written in the form" }}{PARA 256 "
" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+omega^2*x = 0;" "
6#/,&*(%\"dG\"\"#%\"xG\"\"\"*&F&F)*$%\"tGF'F)!\"\"F)*&%&omegaGF'F(F)F)
\"\"!" }{TEXT -1 14 "  ------- (iv)" }}{PARA 0 "" 0 "" {TEXT -1 6 "whe
re " }{XPPEDIT 18 0 "omega^2 = k/M;" "6#/*$%&omegaG\"\"#*&%\"kG\"\"\"%
\"MG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 80 "The solution of (iv) can be determined from the ro
ots of the auxiliary equation " }{XPPEDIT 18 0 "m^2+omega^2=0" "6#/,&*
$%\"mG\"\"#\"\"\"*$%&omegaGF'F(\"\"!" }{TEXT -1 12 ", which are " }
{XPPEDIT 18 0 "omega*i;" "6#*&%&omegaG\"\"\"%\"iGF%" }{TEXT -1 5 " and
 " }{XPPEDIT 18 0 "-omega*i;" "6#,$*&%&omegaG\"\"\"%\"iGF&!\"\"" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 35 "Thus (iv) has the genera
l solution:" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "x(t) \+
= C[1]*cos*omega*t+C[2]*sin*omega*t;" "6#/-%\"xG6#%\"tG,&**&%\"CG6#\"
\"\"F-%$cosGF-%&omegaGF-F'F-F-**&F+6#\"\"#F-%$sinGF-F/F-F'F-F-" }
{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 308 16 "____
____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 53 "The solution can be written in the alternative \+
form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t)=A*sin(
omega*t+phi)" "6#/-%\"xG6#%\"tG*&%\"AG\"\"\"-%$sinG6#,&*&%&omegaGF*F'F
*F*%$phiGF*F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 26 "which i
s a sine wave with " }{TEXT 259 9 "amplitude" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "A=sqrt(C[1]^2+C[2]^2)" "6#/%\"AG-%%sqrtG6#,&*$&%\"CG6#
\"\"\"\"\"#F-*$&F+6#F.F.F-" }{TEXT -1 5 " and " }{TEXT 259 11 "phase a
ngle" }{TEXT -1 1 " " }{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 27 " d
efined by the equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "sin*phi=C[1]/A" "6#/*&%$sinG\"\"\"%$phiGF&*&&%\"CG6#F&F
&%\"AG!\"\"" }{TEXT -1 4 ",   " }{XPPEDIT 18 0 "cos*phi=C[2]/A" "6#/*&
%$cosG\"\"\"%$phiGF&*&&%\"CG6#\"\"#F&%\"AG!\"\"" }{TEXT -1 8 "  and   \+
" }{XPPEDIT 18 0 "tan*phi = C[1]/C[2]" "6#/*&%$tanG\"\"\"%$phiGF&*&&%
\"CG6#F&F&&F*6#\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 47 "This means that the motion is oscillatory with " }{TEXT 259 6 "
period" }{TEXT -1 1 " " }{XPPEDIT 18 0 "T = 2*Pi/omega;" "6#/%\"TG*(\"
\"#\"\"\"%#PiGF'%&omegaG!\"\"" }{TEXT -1 5 " and " }{TEXT 259 9 "frequ
ency" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "f = 1/T;" "6#/%\"fG*&\"\"\"F&%
\"TG!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "omega/(2*Pi);" "6#*&%&ome
gaG\"\"\"*&\"\"#F%%#PiGF%!\"\"" }{TEXT -1 38 ", measured in cycles per
 unit of time." }}{PARA 0 "" 0 "" {TEXT -1 13 "The constant " }
{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 15 " is called the " }
{TEXT 259 27 "(natural) angular frequency" }{TEXT -1 63 " of the motio
n, and it is measured in radians per unit of time." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 29 ": If \+
the weight is suspended " }{TEXT 259 10 "vertically" }{TEXT -1 41 " by
 the spring from a rigid support, the " }{TEXT 259 26 "same differenti
al equation" }{TEXT -1 26 " governs the motion where " }{TEXT 264 1 "x
" }{TEXT -1 120 " is the displacement downwards from an equilibrium po
sition in which the weight would hang without any motion occurring." }
}{PARA 0 "" 0 "" {TEXT -1 74 "The equilibrium position will occur with
 the spring extended by an amount " }{TEXT 271 1 "s" }{TEXT -1 52 ", s
ay, due to the weight acting as a downward force." }}{PARA 0 "" 0 "" 
{TEXT -1 99 "This initially introduces two extra terms in equation (i)
 so that the net force acting downwards is" }}{PARA 256 "" 0 "" {TEXT 
-1 2 "  " }{XPPEDIT 18 0 "F = -k*x-k*s+M*g;" "6#/%\"FG,(*&%\"kG\"\"\"%
\"xGF(!\"\"*&F'F(%\"sGF(F**&%\"MGF(%\"gGF(F(" }{TEXT -1 1 "," }}{PARA 
0 "" 0 "" {TEXT -1 69 "where g is the acceleration downwards due to gr
avity. However, since " }{TEXT 270 1 "s" }{TEXT -1 41 " is the extensi
on at equilibrium, we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "M*g = k*s;" "6#/*&%\"MG\"\"\"%\"gGF&*&%\"kGF&%\"sGF&" }
{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 36 "so these two extra term
s cancel out." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 
0 "" {TEXT -1 30 "Simple harmonic motion example" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT 258 8 "Question" 
}{TEXT 262 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 45 "Solve and interpret t
he initial value problem" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "d^2*x/(d*t^2)+16*x = 0;" "6#/,&*(%\"dG\"\"#%\"xG\"\"\"*
&F&F)*$%\"tGF'F)!\"\"F)*&\"#;F)F(F)F)\"\"!" }{TEXT -1 3 " , " }{TEXT 
279 1 "x" }{TEXT -1 11 "(0) = 10,  " }{XPPEDIT 18 0 "eval(dx/dt,t = 0)
 = 0;" "6#/-%%evalG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!F." }{TEXT -1 
3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 261 
8 "Solution" }{TEXT 263 2 ": " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 27 "The auxiliary equation is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "m^2+16 = 0;" "6#/,&*$%\"mG\"\"#\"\"\"\"#;F(\"\"!" }
{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 24 "which has the solutions
 " }{XPPEDIT 18 0 "m=``" "6#/%\"mG%!G" }{TEXT 282 1 "+" }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "4*i;" "6#*&\"\"%\"\"\"%\"iGF%" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 16 "general solution" }
{TEXT -1 47 " for the homogeneous differential equation is: " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = C[1]*cos*4*t+C[2]*si
n*4*t;" "6#/-%\"xG6#%\"tG,&**&%\"CG6#\"\"\"F-%$cosGF-\"\"%F-F'F-F-**&F
+6#\"\"#F-%$sinGF-F/F-F'F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "x(
0) = 10;" "6#/-%\"xG6#\"\"!\"#5" }{TEXT -1 26 ", we see immediately th
at " }{XPPEDIT 18 0 "C[1]=10" "6#/&%\"CG6#\"\"\"\"#5" }{TEXT -1 2 ". \+
" }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "x(t)=10*cos*4*t+C[2]*sin*4*t" "6#/-%\"xG6#%\"tG,
&**\"#5\"\"\"%$cosGF+\"\"%F+F'F+F+**&%\"CG6#\"\"#F+%$sinGF+F-F+F'F+F+
" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "we have " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt=-40*sin*4*t+4*C[2]*co
s*4*t" "6#/*&%#dxG\"\"\"%#dtG!\"\",&**\"#SF&%$sinGF&\"\"%F&%\"tGF&F(*,
F-F&&%\"CG6#\"\"#F&%$cosGF&F-F&F.F&F&" }{TEXT -1 2 ". " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 
0 "Eval(dx/dt,t=0)=0" "6#/-%%EvalG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!
F." }{TEXT -1 18 ", it follows that " }{XPPEDIT 18 0 "0=4*C[2]" "6#/\"
\"!*&\"\"%\"\"\"&%\"CG6#\"\"#F'" }{TEXT -1 10 ", so that " }{XPPEDIT 
18 0 "C[2]=0" "6#/&%\"CG6#\"\"#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 19 "pa
rticular solution" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "x(t)=10*cos*4*t" "6#/-%\"xG6#%\"tG**\"#5\"\"\"%$cosG
F*\"\"%F*F'F*" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{TEXT 283 9 "_________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 39 
"satisfies the given initial condtions. " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 390 "The given second order differenti
al equation models a system consisting of a mass resting on a horizont
al frictionless surface and attached to a horizontal spring which has \+
the other end fixed to a rigid support. If the mass is pulled aside to
 a distance of 10 units to the right of the equilibrium position, and \+
then released from rest, its motion is governed by the initial value p
roblem." }}{PARA 0 "" 0 "" {TEXT -1 155 "The solution shows that, once
 the system is set in motion, the mass oscillates back and forth 10 un
its on either side of the equilibrium position given by " }{XPPEDIT 
18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "The graph of the solution is as
 follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 77 "g := t -> 10*cos(4*t):\n'g(t)'=g(t);\nplot(g(t),t=0..
2*Pi,labels=[`t`,`x(t)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6
#%\"tG,$*&\"#5\"\"\"-%$cosG6#,$*&\"\"%F+F'F+F+F+F+" }}{PARA 13 "" 1 "
" {GLPLOT2D 406 169 169 {PLOTDATA 2 "6%-%'CURVESG6$7[x7$$\"\"!F)$\"#5F
)7$$\"3Hjqb\"[W>r\"!#>$\"3CwUCHJcw**!#<7$$\"3eET6j*))QU$F/$\"3Oq#fVPij
!**F27$$\"3_*=rYWLe8&F/$\"3S>)[DzE(*y*F27$$\"3;`#Gi#zxZoF/$\"3T0UU)4.s
i*F27$$\"3\"zBM*)omr-\"!#=$\"3/P'\\X8]x;*F27$$\"3i]cC&eb&p8FE$\"3tC<m*
*ygO&)F27$$\"3q5)>63x`'>FE$\"3q'[abjVd1(F27$$\"3YqR*pd)>hDFE$\"3=3GhVv
V&>&F27$$\"3'fph0PIi*GFE$\"3,>80\"y9s+%F27$$\"3Y@%HT;i7B$FE$\"3?/`ZaC8
ZFF27$$\"3'p9(pdRHmNFE$\"3.\"Hl$[qyP9F27$$\"3Zs[E^dK,RFE$\"3z.H\\'f%eE
5FE7$$\"3CVu.`$y&QUFE$!3O.1[se7V7F27$$\"3+9+\"[&4$ed%FE$!3%Gspcs@jc#F2
7$$\"3y%e#ecN38\\FE$!3Mv%GEp&)G%QF27$$\"3ab^NehL]_FE$!3u$R>0q@'\\]F27$
$\"3]9<Qde)fe&FE$!3o^rQ?GbfhF27$$\"3Ls#3kbN;#fFE$!35(yt\"y-ierF27$$\"3
;I[Vb_GdiFE$!3XGy//A%)G!)F27$$\"35*QhW&\\$Hf'FE$!3_([pVxbXv)F27$$\"3J-
7h\">DT!pFE$!3Wd!=VLfoG*F27$$\"3]:5wGaJ:sFE$!3S3:WC_Xv'*F27$$\"3c@fLZ0
\"4P(FE$!3E)Gs'**R*Q\")*F27$$\"3fF3\"fm0l_(FE$!3oK)=W>IV\"**F27$$\"3uM
d[%y+@o(FE$!3wZ3;()[Pw**F27$$\"3zS11.fpPyFE$!3m?_s?yy****F27$$\"3'y&R#
fj0))*zFE$!3)[.)f)\\DK)**F27$$\"3'QF(yo`\"*f\")FE$!3Y&oYW.<_#**F27$$\"
3%)*e];5D5K)FE$!39*H]\\D.g#)*F27$$\"3#e!R^M[8#[)FE$!3%yrzQ0'*fo*F27$$
\"3yP0C+VN/))FE$!3=HLSML4'G*F27$$\"3upr'fwtl7*FE$!3Cv7@$*G9K()F27$$\"3
#=Oh$)f8)f%*FE$!3i#>AnZ/q+)F27$$\"3!Rbb2V`Iz*FE$!3^%4UHo4)RrF27$$\"3gu
\\JE$HE,\"F2$!3#o=L*fQ%f9'F27$$\"3!QRa&4L&f/\"F2$!3h&H@:pRI/&F27$$\"31
OpNA/<z5F2$!35j\"\\i>4Y&QF27$$\"3_y%f^`(Q76F2$!3!zq7eQH#)f#F27$$\"3+@?
'zk/c9\"F2$!3#HYT'4y/'H\"F27$$\"3YjXwg<#)y6F2$\"3I'Qtohe!)*GF/7$$\"38S
T=W'*)H@\"F2$\"3)e'>b\"4p6R\"F27$$\"3!or.w_drC\"F2$\"3Ry3=BATFFF27$$\"
3W$HB5TD8G\"F2$\"3\\?_Az*)y7SF27$$\"36qGW%H$\\:8F2$\"3tagK=qKB_F27$$\"
3SH22vMov8F2$\"3<m;7/+'f5(F27$$\"3$*)e)pbO(eV\"F2$\"3)p*[?1nny&)F27$$
\"3Q>$*Q*f`(p9F2$\"3c#H**Re(G%>*F27$$\"3/]+3VNj.:F2$\"3zH[LU!)HT'*F27$
$\"3P:a#\\^t0_\"F2$\"3-8W!3D\"*))z*F27$$\"3q!yqn[8v`\"F2$\"3zLNi=5^6**
F27$$\"3/YhheMXa:F2$\"3s'>T4YS'y**F27$$\"396:YIMRr:F2$\"3Kl/Rx9(*****F
27$$\"3[K)e%fHS)e\"F2$\"3is@Wn2@v**F27$$\"3\"Q:c%)[7ag\"F2$\"3Gh#\\*p2
H/**F27$$\"39vMX<?UA;F2$\"3%=])3g'Rvy*F27$$\"3Z'z]kaJ%R;F2$\"3e_L\">p(
\\D'*F27$$\"39RaW/1Xt;F2$\"3'o/$e@muo\"*F27$$\"3!=3SCmpuq\"F2$\"301>td
?[U&)F27$$\"3%z2'RGLDS<F2$\"3]xDxzhO*y(F27$$\"33u?N%*p.t<F2$\"3Sl_(\\'
Q\\-pF27$$\"3?q!3.m?e!=F2$\"3%=AZL>%4(*eF27$$\"3LmSEEVgQ=F2$\"3Kg\\sd:
V!z%F27$$\"35QMoN]Po=F2$\"3s!ftGqvOr$F27$$\"3')4G5Xd9)*=F2$\"3WWL$yW>V
e#F27$$\"3j\"=AXX;z#>F2$\"3w^(RM'*e$=9F27$$\"3S`:%R;(od>F2$\"3;K$H#=))
3BBFE7$$\"3M]e#pT(3$*>F2$!3qf?&Q==4=\"F27$$\"3^Z,\"*pw[G?F2$!3SA%4\"yj
]qDF27$$\"3oWW*G#z)Q1#F2$!3RX\\d0)R'3RF27$$\"3#=uye<)G*4#F2$!3Ug+xfD`o
^F27$$\"3!zz^8\\l#f@F2$!3/z=W&GfY0(F27$$\"3Ua[#o!GC>AF2$!3.<VYI')oO&)F
27$$\"3W_![\\uETD#F2$!3)[`[ra'3y\"*F27$$\"3W]72$o5!*G#F2$!3F<*Gvzu5k*F
27$$\"3A\\G8_EX1BF2$!3R#p?p57F!)*F27$$\"3,[W>@Y*QK#F2$!3MmeJ7Nl;**F27$
$\"3zYgD!fO8M#F2$!3mV>TAYM#)**F27$$\"3-YwJf&y(eBF2$!3_skp6eY****F27$$
\"3;#3ltl<UP#F2$!35'3a))[CS(**F27$$\"3w=DTbnl*Q#F2$!3bcL?,eb5**F27$$\"
3\"\\&*fM&e40CF2$!3x)3-?t,$4)*F27$$\"31\"R2:&\\`?CF2$!3W.bcE$[1n*F27$$
\"3OjAgZJT^CF2$!3)=QyPi)R$G*F27$$\"3oNrpV8H#[#F2$!3cvN`Qoqa()F27$$\"3X
@Su='ph^#F2$!3z%*3xj;b@!)F27$$\"3C24z$*y/]DF2$!3uq28.cJTrF27$$\"3Y$zP)
oh#Re#F2$!3WB?B9%Q,8'F27$$\"3EzY)QW/yh#F2$!3\"on9Rvgl+&F27$$\"3+yMCtX1
]EF2$!39&4E^J)*4&QF27$$\"3=xAg-ZK#o#F2$!3SYz5#3+9j#F27$$\"3Ow5'>$[e9FF
2$!3+#yJe(f/o8F27$$\"35v)>8'\\%ou#F2$!3iV49x0N%>)F/7$$\"3Exf*>-00y#F2$
\"3Kv*GQA\"4h7F27$$\"3Vz?n#3lT\"GF2$\"3)p<[ET*H\"e#F27$$\"3g\"=[L9Dy%G
F2$\"3;9F#eY%yaQF27$$\"3w$GCS?&[\")GF2$\"3Mx)*\\1b\\e]F27$$\"3*Q3$\\!4
1L%HF2$\"37E*[,&)og,(F27$$\"3Z%)='p(p70IF2$\"3)4-S&*H(zY&)F27$$\"3g`,t
a\"4=2$F2$\"30KnaJS%Hh*F27$$\"3sA%)\\K8\\QJF2$\"3%3qmWbI#****F27$$\"3E
pD+Vt!e:$F2$\"3saH#GwRQ)**F27$$\"3#er1NNBJ<$F2$\"3&4l\")3dq0#**F27$$\"
3Oi3,k$R/>$F2$\"3)*o`t\"RE(4)*F27$$\"3#*3]^u`v2KF2$\"3Ub3g)yQ=l*F27$$
\"3--L_&R(QUKF2$\"3g!Q5q$R=)>*F27$$\"3c&fJlT>qF$F2$\"3_olW`xHo&)F27$$
\"3e!=@(oRJPLF2$\"3!=+?SP9))3(F27$$\"39l2\"4_3wR$F2$\"3!pZ,;D\"**)>&F2
7$$\"3%yv/]So,V$F2$\"3[!G='G/cXSF27$$\"3_]()4*GGFY$F2$\"3*GOw9R/O#GF27
$$\"3?VF>t\")G&\\$F2$\"3+N_6)3@Qb\"F27$$\"3MOnGd![y_$F2$\"3sS@!e#e=xDF
E7$$\"3?#yUNg&[hNF2$!31)=^x#4h&3\"F27$$\"3iF))z\\J7&f$F2$!35q*[#RkJ4CF
27$$\"31t[0'pg(GOF2$!3!ywmm/q%*o$F27$$\"3#*=4JU#)RiOF2$!3A8F')36$H!\\F
27$$\"3GUz%)=gI&p$F2$!3bcK$eFZW+'F27$$\"36m\\Q&z8#GPF2$!36b%p*Hg2-qF27
$$\"3[*)>#>d@6w$F2$!3sSuU`mbyyF27$$\"3%G,f%[$HSz$F2$!3ta6P-Xs=')F27$$
\"3eS3B!G4x&QF2$!3rBgHSnX='*F27$$\"3KoE+7#*Q@RF2$!35*eZn')*[(***F27$$
\"3GM!y(Qc1RRF2$!3aGSkh\"Q$))**F27$$\"3o+Mbl?ucRF2$!3=ru3(es#H**F27$$
\"35n(GB\\=W(RF2$!3#*3/V1$)e?)*F27$$\"31LT5>\\4#*RF2$!33(p[$Q%GGm*F27$
$\"3)\\'[lsxWFSF2$!3$*)*[A\"owM?*F27$$\"3y(f0ii+G1%F2$!3C&*>(>w(Qg&)F2
7$$\"378E]Pnc%4%F2$!3n'QfVZIj$yF27$$\"3ZG'*z[GLETF2$!31x!*eTG#f)pF27$$
\"3#Qk'4g*)4eTF2$!3Gr<.\\l(G-'F27$$\"3;fORr]')*=%F2$!3%)[\\vH%>F'\\F27
$$\"3))=Rav@yBUF2$!38>iP&)*)*Gu$F27$$\"3sxTpz#*pdUF2$!3c[UEZEHaCF27$$
\"3cOW%QQ;;H%F2$!3sqtnU?e?6F27$$\"3R&p%*z[LbK%F2$\"3'>$\\=#)GAPBFE7$$
\"3i&pFaUniN%F2$\"31x\")=Rm)yX\"F27$$\"3'epgGO,qQ%F2$\"3m?4V&RX+m#F27$
$\"35'p$H+`t<WF2$\"3QYJSjH0AQF27$$\"3M'pExBp%[WF2$\"3`?ne)zpj#\\F27$$
\"3oPwB\\&p?[%F2$\"3Y/#Gk-J![gF27$$\"3,z&[2')pc^%F2$\"3e89TAoggqF27$$
\"3O?&fA<q#\\XF2$\"3AP&fI%Q$e%zF27$$\"3oh/x$[qGe%F2$\"3U8Thtcu(o)F27$$
\"3>](o$ed[9YF2$\"3zj$QtbYHC*F27$$\"3qQq'H.,hk%F2$\"3Al-UYt_]'*F27$$\"
3Q$=m-n3>m%F2$\"33kjBx2$oz*F27$$\"33G`c2jrxYF2$\"3ZRP>C&yR!**F27$$\"3)
=Zk[%R_$p%F2$\"3w=CMUBar**F27$$\"3e;O;#eJ$4ZF2$\"3iB)4Y>_#****F27$$\"3
ns[!z6bes%F2$\"3IU)4%el\\&)**F27$$\"3lHhk`'yBu%F2$\"3Sj``*zN\"G**F27$$
\"3k'Q(Q*=-*eZF2$\"3fG%pNS?u#)*F27$$\"3sU'G^sDax%F2$\"35-b5z,z$o*F27$$
\"3zb6h'zs%3[F2$\"3wCWH_(y/F*F27$$\"3&)oO4o)>:%[F2$\"3gevU1JT&p)F27$$
\"3WXD#4UHQ([F2$\"3WU')>\\cT')zF27$$\"3;@9vt*Qh!\\F2$\"3o+b@E0@WrF27$$
\"3w(H!eE&[%Q\\F2$\"3?4B&G6XG='F27$$\"3Ou\"4%z!e2(\\F2$\"34z`P#Ha$=^F2
7$$\"3/&\\Fp=$e/]F2$\"3(4py>IwF\"RF27$$\"3t:eW%H3%Q]F2$\"3b`aJC%zcj#F2
7$$\"3IPT'>SLA2&F2$\"3xk')Q:oS58F27$$\"3*zX#[4&eg5&F2$!3A3x_Z)z<)QF/7$
$\"3Y!*>B3hjQ^F2$!3()y->=!)*yL\"F27$$\"3#Q_\")pq87<&F2$!36\"zC1z\"H9EF
27$$\"3>d5t08z._F2$!3iS*)o#ebj%QF27$$\"3n*e![/*ojB&F2$!3mZ0=bv>8]F27$$
\"3-J))[/Uop_F2$!31\\pz]()R=hF27$$\"3Dtq\\/&**HI&F2$!35N`Srn5:rF27$$\"
3[:`]/[JO`F2$!3O)oG)3xk&)zF27$$\"3#ob8X5I'p`F2$!3L3%\\_)[e9()F27$$\"3@
)><q`pES&F2$!3))Qi*zT+\\G*F27$$\"3fR3_p*3dV&F2$!3')ox,x[G$p*F27$$\"3%)
fEx&oGAX&F2$!3Dj8KkJTM)*F27$$\"3)4[C?S[(oaF2$!35W(G8$fhK**F27$$\"37-jF
=\"o_[&F2$!3+L#p)RXY()**F27$$\"3PA\"GX$yy,bF2$!3g+(4_e>()***F27$$\"3x&
=#4Qx'p^&F2$!3-i3K=Geq**F27$$\"31]ilTw9KbF2$!3P\\g0'3(p0**F27$$\"3O9.A
XvKZbF2$!3F-bGS:I/)*F27$$\"3wxVy[u]ibF2$!3114_)*)pnm*F27$$\"3W0D\"fDnG
f&F2$!3ky_\")psX&G*F27$$\"39L1/jqABcF2$!3gjB:&pvtw)F27$$\"3CW>6GG-ecF2
$!3AI]0JSK:!)F27$$\"3WaK=$f=Gp&F2$!3G!*=@:1D3rF27$$\"3blXDeVhFdF2$!3Y-
,*o&))pjgF27$$\"3mweKB,TidF2$!34oL'))3r=!\\F27$$\"3w$)*3/(=`$z&F2$!3[H
BLM#)o\"y$F27$$\"3w\"4#\\<OlCeF2$!3CE%yE;wHg#F27$$\"3w*>vXOvd&eF2$!3kV
<Y4y(RQ\"F27$$\"3'oIe;6(*o)eF2$!3F2\"QK@'fN9FE7$$\"3(y8!=&Qz+#fF2$\"31
>+H7R'4=\"F27$$\"3yp>qe;E`fF2$\"3eKNk:Hr%[#F27$$\"3o,QAKRW')fF2$\"3jy4
wDHvWPF27$$\"3nKcu0ii>gF2$\"37^(\\<P=*Q\\F27$$\"3)Rsoa(eQ^gF2$\"3^%[]2
!*z2+'F27$$\"3G:=>Xb9$3'F2$\"35p:>lg#f'pF27$$\"3Y2\\\"\\@0\\6'F2$\"3y&
o\"\\w9!)=yF27$$\"3w)*zj%)[mYhF2$\"3_x)p'G-mX&)F27$$\"31*\\Lr)\\z!='F2
$\"3_hVgBh&H<*F27$$\"3P***G'*3D\\@'F2$\"3K*))*)*fMbH'*F27$$\"3'*\\n(39
!*>B'F2$\"3?9$*G.h0\"z*F27$$\"3o*\\C@>b!\\iF2$\"3>C>(oYbp!**F27$$\"3Q
\\APV-7miF2$\"3/?XXK<rw**F27$$\"3)****>YH&=$G'F2$\"2w)**************!#
;-%'COLOURG6&%$RGBG$F+!\"\"F(F(-%+AXESLABELSG6$%\"tG%%x(t)G-%%VIEWG6$;
F($\"+3`=$G'!\"*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 15 "Solution using " }{TEXT 0 6 "dsolve" }{TEXT -1 5 " and " }
{TEXT 0 7 "desolve" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "de := diff(x(t),t$2)+16*x(t)
=0;\nic := x(0)=10,D(x)(0)=0;\ndsolve(\{de,ic\},x(t));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"
\"\"*&\"#;F2F*F2F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%
\"xG6#\"\"!\"#5/--%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%
\"xG6#%\"tG,$*&\"#5\"\"\"-%$cosG6#,$*&\"\"%F+F'F+F+F+F+" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }
{TEXT 0 7 "desolve" }{TEXT -1 24 " can be used instead of " }{TEXT 0 
6 "dsolve" }{TEXT -1 45 " if we want to see the steps in the solution.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 89 "de := diff(x(t),t$2)+16*x(t)=0;\nic := x(0)=10,D(x)(0)=0;\ndes
olve(\{de,ic\},x(t),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
deG/,&-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&\"#;F2F*F2F2\"\"!
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"!\"#5/--%\"DG
6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8auxiliary~equation~.~.~G
/,&*$)%\"mG\"\"#\"\"\"F*\"#;F*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+roots~.~.~G^#\"\"%^#!\"%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#%6general~solution~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#
%\"tG,&*&&%\"CG6#\"\"\"F--%$sinG6#,$*&\"\"%F-F'F-F-F-F-*&&F+6#\"\"#F--
%$cosGF0F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%Afrom~the~initial~conditions~.~.~G" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/\"\"!,$*&\"\"%\"\"\"&%\"CG6#F(F(F(" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/\"#5&%\"CG6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-so~that~.~.~G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%\"CG6#\"\"#\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%
!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,$*&\"#5\"\"\"-%$c
osG6#,$*&\"\"%F+F'F+F+F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Damped mot
ion" }{TEXT 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 419 95 95 {PLOTDATA 2 "6)-
%'CURVESG6%7D7$\"\"!$\"\"&!\"\"7$$F*!\"#F)7$$\"\"\"F+$\"\")F+7$$\"\"#F
+F57$$\"\"$F+F27$$\"\"%F+F57$F)F27$$\"\"'F+F57$$\"\"(F+F27$F2F57$$\"\"
*F+F27$F1F57$$\"#6F+F27$$\"#7F+F57$$\"#8F+F27$$\"#9F+F57$$\"#:F+F27$$
\"#;F+F57$$\"#<F+F27$$\"#=F+F57$$\"#>F+F27$F6F57$$\"#@F+F27$$\"#AF+F57
$$\"#BF+F27$$\"#CF+F57$$\"#DF+F27$$\"#EF+F57$$\"#FF+F27$$\"#GF+F57$$\"
#HF+F27$F9F57$$\"$0$F.F)7$$\"#JF+F)-%*THICKNESSG6#F6-%&COLORG6&%$RGBGF
(F?F(-F$6%7%7$F(F17$F(F(7$F*F(Faq-Feq6&FgqF5F5F5-%)POLYGONSG6$7'7$F_q$
F6F.7$F_q$\"#)*F.7$$\"#OF+Fgr7$FjrFerFdr-Feq6&FgqF2F2F2-Far6$7'7$$\"#X
F+F(7$FcsF17$F*F1F]rFbs-Feq6&FgqF2F)F(-F$6&7%7$F<F1FfsF]r7$7$FjrF)7$Fc
sF)7$FbsFesFaq-%%TEXTG6$7$$\"$N$F.F)%\"MG-%*AXESSTYLEG6#%%NONEG" 1 2 
0 1 10 0 2 9 1 1 2 1.000000 45.000000 44.000000 0 0 "Curve 1" "Curve 2
" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "Suppose that the mass considered \+
in the first section is connected to a dashpot damping device as well \+
as to the spring." }}{PARA 0 "" 0 "" {TEXT -1 88 "Suppose also that th
e damping force exerted on the mass is proportional to the velocity." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 29 "Newton
's second law of motion" }{TEXT -1 13 " implies that" }}{PARA 256 "" 
0 "" {TEXT -1 2 "  " }{TEXT 265 1 "M" }{TEXT -1 1 " " }{XPPEDIT 18 0 "
d^2*x/(d*t^2) = -k*x-beta;" "6#/*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&
F(!\"\",&*&%\"kGF(F'F(F,%%betaGF," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/
dt;" "6#*&%#dxG\"\"\"%#dtG!\"\"" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" 
{TEXT -1 6 "where " }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 8 " i
s the " }{TEXT 259 16 "damping constant" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "The motion of the ma
ss is determined by the differential equation" }}{PARA 256 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+beta/M;" "6#,&*(%\"dG\"
\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(*&%%betaGF(%\"MGF,F(" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "dx/dt+k/M;" "6#,&*&%#dxG\"\"\"%#dtG!\"\"F&*&%\"k
GF&%\"MGF(F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" 
}{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+2*lambda;" "6#,&*(%\"dG\"
\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(*&F&F(%'lambdaGF(F(" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "dx/dt+omega^2*x = 0;" "6#/,&*&%#dxG\"\"\"%#dtG!
\"\"F'*&%&omegaG\"\"#%\"xGF'F'\"\"!" }{TEXT -1 3 ",  " }}{PARA 0 "" 0 
"" {TEXT -1 7 " where " }{XPPEDIT 18 0 "2*lambda = beta/M;" "6#/*&\"\"
#\"\"\"%'lambdaGF&*&%%betaGF&%\"MG!\"\"" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "omega^2 = k/M;" "6#/*$%&omegaG\"\"#*&%\"kG\"\"\"%\"MG!
\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 80 "This differential equation is linear, homogenous, and h
as constant coefficients." }}{PARA 0 "" 0 "" {TEXT -1 35 "The auxiliar
y equation has the form" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "m^2+2*lambda*m+omega^2 = 0;" "6#/,(*$%\"mG\"\"#\"\"\"*(
F'F(%'lambdaGF(F&F(F(*$%&omegaGF'F(\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "
" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 20 " \+
are both positive. " }}{PARA 0 "" 0 "" {TEXT -1 28 "Completing the squ
are gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m^2+2*l
ambda*m+lambda^2 = lambda^2-omega^2;" "6#/,(*$%\"mG\"\"#\"\"\"*(F'F(%'
lambdaGF(F&F(F(*$F*F'F(,&*$F*F'F(*$%&omegaGF'!\"\"" }{TEXT -1 2 ", " }
}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "(m+lambda)^2 = lambda^2-omega^2;" "6#/*$,&%\"mG
\"\"\"%'lambdaGF'\"\"#,&*$F(F)F'*$%&omegaGF)!\"\"" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 41 "The roots of the auxiliary equation are: \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m[1] = -lambda+sq
rt(lambda^2-omega^2);" "6#/&%\"mG6#\"\"\",&%'lambdaG!\"\"-%%sqrtG6#,&*
$F)\"\"#F'*$%&omegaGF0F*F'" }{TEXT -1 6 " and  " }{XPPEDIT 18 0 "m[2] \+
= -lambda-sqrt(lambda^2-omega^2);" "6#/&%\"mG6#\"\"#,&%'lambdaG!\"\"-%
%sqrtG6#,&*$F)F'\"\"\"*$%&omegaGF'F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The nature of the mo
tion is determined by the relative magnitudes of " }{XPPEDIT 18 0 "ome
ga" "6#%&omegaG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "lambda" "6#%'lamb
daG" }{TEXT -1 52 ". We consider three cases in the following sections
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note
" }{TEXT -1 100 ": As in the undamped case, we could just as well cons
ider a system in which the weight is suspended " }{TEXT 259 10 "vertic
ally" }{TEXT -1 22 " from a rigid support." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Over-damped systems " 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 258 "" 0 "" 
{TEXT -1 36 "Consider the differential equation: " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+2*lambda;" "6#,&*(%\"dG
\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(*&F&F(%'lambdaGF(F(" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "dx/dt+omega^2*x = 0;" "6#/,&*&%#dxG\"\"\"%#dt
G!\"\"F'*&%&omegaG\"\"#%\"xGF'F'\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 
0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "0<omega" "6#2\"\"!%&omegaG" }
{XPPEDIT 18 0 "`` < lambda;" "6#2%!G%'lambdaG" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 14 "The solutions " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m[1] = -lam
bda+sqrt(lambda^2-omega^2);" "6#/&%\"mG6#\"\"\",&%'lambdaG!\"\"-%%sqrt
G6#,&*$F)\"\"#F'*$%&omegaGF0F*F'" }{TEXT -1 6 " and  " }{XPPEDIT 18 0 
"m[2] = -lambda-sqrt(lambda^2-omega^2);" "6#/&%\"mG6#\"\"#,&%'lambdaG!
\"\"-%%sqrtG6#,&*$F)F'\"\"\"*$%&omegaGF'F*F*" }{TEXT -1 1 " " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "of the auxiliar
y equation are " }{TEXT 259 17 "real and distinct" }{TEXT -1 63 ", so \+
the general solution of the differential equation above is" }}{PARA 
256 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 18 0 "x(t) = C[1]*exp(m[1]*t)+C
[2]*exp(m[2]*t);" "6#/-%\"xG6#%\"tG,&*&&%\"CG6#\"\"\"F--%$expG6#*&&%\"
mG6#F-F-F'F-F-F-*&&F+6#\"\"#F--F/6#*&&F36#F8F-F'F-F-F-" }{TEXT -1 2 ",
 " }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT -1 3 " \+
  " }{XPPEDIT 18 0 "x(t) = C[1]*exp((-lambda+sqrt(lambda^2-omega^2))*t
)+C[2]*exp((-lambda-sqrt(lambda^2-omega^2))*t);" "6#/-%\"xG6#%\"tG,&*&
&%\"CG6#\"\"\"F--%$expG6#*&,&%'lambdaG!\"\"-%%sqrtG6#,&*$F3\"\"#F-*$%&
omegaGF:F4F-F-F'F-F-F-*&&F+6#F:F--F/6#*&,&F3F4-F66#,&*$F3F:F-*$F<F:F4F
4F-F'F-F-F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "sqrt(lambda^2-omega^2)<l
ambda" "6#2-%%sqrtG6#,&*$%'lambdaG\"\"#\"\"\"*$%&omegaGF*!\"\"F)" }
{TEXT -1 18 ", it follows that " }{XPPEDIT 18 0 "m[1] = -lambda+sqrt(l
ambda^2-omega^2);" "6#/&%\"mG6#\"\"\",&%'lambdaG!\"\"-%%sqrtG6#,&*$F)
\"\"#F'*$%&omegaGF0F*F'" }{TEXT -1 6 " and  " }{XPPEDIT 18 0 "m[2] = -
lambda-sqrt(lambda^2-omega^2);" "6#/&%\"mG6#\"\"#,&%'lambdaG!\"\"-%%sq
rtG6#,&*$F)F'\"\"\"*$%&omegaGF'F*F*" }{TEXT -1 22 " , are both negativ
e. " }}{PARA 0 "" 0 "" {TEXT -1 17 "Thus we see that " }{XPPEDIT 18 0 
"x(t)->0" "6#f*6#-%\"xG6#%\"tG7\"6$%)operatorG%&arrowG6\"\"\"!F-F-F-" 
}{TEXT -1 4 " as " }{XPPEDIT 18 0 "t->infinity" "6#f*6#%\"tG7\"6$%)ope
ratorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 34 "Consider the
 initial value problem" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "d^2*x/(d*t^2)+5;" "6#,&*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!
\"\"F(\"\"&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+4*x = 0;" "6#/,&*
&%#dxG\"\"\"%#dtG!\"\"F'*&\"\"%F'%\"xGF'F'\"\"!" }{TEXT -1 3 ",  " }
{XPPEDIT 18 0 "x(0) = 1" "6#/-%\"xG6#\"\"!\"\"\"" }{TEXT -1 3 ",  " }
{XPPEDIT 18 0 "Eval(dx/dt,t = 0) = 4" "6#/-%%EvalG6$*&%#dxG\"\"\"%#dtG
!\"\"/%\"tG\"\"!\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 26 "The auxiliary equation is " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m^2+5*m+4=0" "6#/,(*$%\"mG
\"\"#\"\"\"*&\"\"&F(F&F(F(\"\"%F(\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 
0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "(m+1)*(m+4)=0" "6#/*&,&%\"mG\"\"\"F'F'F',&F&F'\"\"%F'F'\"\"!" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "The solutions " }
{XPPEDIT 18 0 "m=-1" "6#/%\"mG,$\"\"\"!\"\"" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "m=-4" "6#/%\"mG,$\"\"%!\"\"" }{TEXT -1 76 " show that t
he general solution of the homogeneous differential equation is " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t)=C[1]*exp(-t)+C[2
]*exp(-4*t)" "6#/-%\"xG6#%\"tG,&*&&%\"CG6#\"\"\"F--%$expG6#,$F'!\"\"F-
F-*&&F+6#\"\"#F--F/6#,$*&\"\"%F-F'F-F2F-F-" }{TEXT -1 2 ". " }}{PARA 
0 "" 0 "" {TEXT -1 6 "Given " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "x = C[1]*exp(-t)+C[2]*exp(-4*t)" "6#/%\"xG,&*&&%\"CG6#
\"\"\"F*-%$expG6#,$%\"tG!\"\"F*F**&&F(6#\"\"#F*-F,6#,$*&\"\"%F*F/F*F0F
*F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "we have " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt = -C[1]*exp(-t)-4*C[2
]*exp(-4*t);" "6#/*&%#dxG\"\"\"%#dtG!\"\",&*&&%\"CG6#F&F&-%$expG6#,$%
\"tGF(F&F(*(\"\"%F&&F,6#\"\"#F&-F/6#,$*&F4F&F2F&F(F&F(" }{TEXT -1 2 ".
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The \+
initial condition " }{XPPEDIT 18 0 "x(0)=1" "6#/-%\"xG6#\"\"!\"\"\"" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "Eval(dx/dt,t = 0) = 4;" "6#/-%%Eval
G6$*&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!\"\"%" }{TEXT -1 29 " give rise t
o the equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
1=C[1]+C[2]" "6#/\"\"\",&&%\"CG6#F$F$&F'6#\"\"#F$" }{TEXT -1 2 ", " }}
{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "4 = -C[1]-4*C[2];" "6#/\"\"%,&&%\"CG6#\"\"\"!\"\"*&F$F)
&F'6#\"\"#F)F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 "Adding
 these two equations gives " }{XPPEDIT 18 0 "5 = -3*C[2];" "6#/\"\"&,$
*&\"\"$\"\"\"&%\"CG6#\"\"#F(!\"\"" }{TEXT -1 9 " so that " }{XPPEDIT 
18 0 "C[2] = -5/3;" "6#/&%\"CG6#\"\"#,$*&\"\"&\"\"\"\"\"$!\"\"F-" }
{TEXT -1 7 ". Then " }{XPPEDIT 18 0 "C[1] = 8/3;" "6#/&%\"CG6#\"\"\"*&
\"\")F'\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 20 "It \+
follows that the " }{TEXT 259 19 "particular solution" }{TEXT -1 1 " \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = 8/3;" "6#/
-%\"xG6#%\"tG*&\"\")\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "
exp(-t)-5/3;" "6#,&-%$expG6#,$%\"tG!\"\"\"\"\"*&\"\"&F*\"\"$F)F)" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-4*t)" "6#-%$expG6#,$*&\"\"%\"\"\"%
\"tGF)!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{TEXT 289 15 "_______________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 38 "satisfies the given initial condtions." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 56 "We construct the solution and plot its graph as follows.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 88 "g := t -> 8/3*exp(-t)-5/3*exp(-4*t):\n'g(t)'=g(t);\nplot(g(t),
t=0..5,labels=[`t`,`x(t)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"g
G6#%\"tG,&*&#\"\")\"\"$\"\"\"-%$expG6#,$F'!\"\"F-F-*&#\"\"&F,F--F/6#,$
*&\"\"%F-F'F-F2F-F2" }}{PARA 13 "" 1 "" {GLPLOT2D 790 238 238 
{PLOTDATA 2 "6%-%'CURVESG6$7\\o7$$\"\"!F)$\"2y***************!#<7$$\"3
ALL$3FWYs#!#>$\"3+m\"R*>%=/5\"F,7$$\"3WmmmT&)G\\aF0$\"3m+`:YM*\\=\"F,7
$$\"3m****\\7G$R<)F0$\"3wc$eU;3bD\"F,7$$\"3GLLL3x&)*3\"!#=$\"3!oB%**)e
dNJ\"F,7$$\"3))**\\i!R(*Rc\"F@$\"3%*G*z[96!*Q\"F,7$$\"3umm\"H2P\"Q?F@$
\"3\\#)p@EsUP9F,7$$\"3YLek.pu/BF@$\"3SAwy9B\"[X\"F,7$$\"3!***\\PMnNrDF
@$\"3QGX32h:m9F,7$$\"37$eR(\\;m/FF@$\"3Q>78o<zp9F,7$$\"3MmT5ll'z$GF@$
\"3#GIb#zB=s9F,7$$\"37](o/[r7(HF@$\"3!)HW:xlSt9F,7$$\"3MLL$eRwX5$F@$\"
3IZ`2I(QNZ\"F,7$$\"3:L$3F%\\wQKF@$\"3ouk%p+SEZ\"F,7$$\"3_LLe*[`HP$F@$
\"3'z5,!zPxq9F,7$$\"3*QLek.Ur]$F@$\"3=$3%[IT+o9F,7$$\"3rLLL$eI8k$F@$\"
331tL;;Rk9F,7$$\"3*QL$3xwq4RF@$\"3_-2ovO'[X\"F,7$$\"33ML$3x%3yTF@$\"3#
p@u(p1hU9F,7$$\"3h+]PfyG7ZF@$\"3!zlUJ3c:T\"F,7$$\"3emm\"z%4\\Y_F@$\"3g
\"Rzh*Qlt8F,7$$\"3`LLeR-/PiF@$\"3G;b:kap\"H\"F,7$$\"3]***\\il'pisF@$\"
3Klsg.El)>\"F,7$$\"3>MLe*)>VB$)F@$\"3E)ork,%Q+6F,7$$\"3Y++DJbw!Q*F@$\"
35%>G%3Qd/5F,7$$\"3%ommTIOo/\"F,$\"3*fN>^\\L\"3\"*F@7$$\"3YLL3_>jU6F,$
\"3\")o!*Gq&[NL)F@7$$\"37++]i^Z]7F,$\"3q].I)f7W_(F@7$$\"33++](=h(e8F,$
\"3=Z!30V(3!y'F@7$$\"3/++]P[6j9F,$\"3#4:l6$o#e7'F@7$$\"3UL$e*[z(yb\"F,
$\"3!pK'okxv#e&F@7$$\"3wmm;a/cq;F,$\"3iU&GvLEi*\\F@7$$\"3%ommmJ<gw\"F,
$\"3'pzP)fW3YXF@7$$\"3/+]iSj0x=F,$\"3g^$*QED#>2%F@7$$\"3gmmm\"pW`(>F,$
\"3'[!Q1re&Gp$F@7$$\"3K+]i!f#=$3#F,$\"3\"\\g-E*y(oJ$F@7$$\"3?+](=xpe=#
F,$\"3-M%obEST*HF@7$$\"37nm\"H28IH#F,$\"3;]l\\\"*Ge!p#F@7$$\"3um;zpSS
\"R#F,$\"3>L@Ae:')QCF@7$$\"3GLL3_?`(\\#F,$\"3!*4Q]pzd$>#F@7$$\"3fL$e*)
>pxg#F,$\"3a_*3oy3['>F@7$$\"33+]Pf4t.FF,$\"3!31*>\"3Q^y\"F@7$$\"3uLLe*
Gst!GF,$\"3Ti'[;Vp%4;F@7$$\"30+++DRW9HF,$\"3'yE*fK15Y9F@7$$\"3:++DJE>>
IF,$\"3zb\\=$4ABI\"F@7$$\"3F+]i!RU07$F,$\"3-3'RyvAo<\"F@7$$\"3+++v=S2L
KF,$\"3CqyVA')e^5F@7$$\"3Jmmm\"p)=MLF,$\"3%Rr;:GiY]*F07$$\"3B++](=]@W$
F,$\"3:Lg^yC/K&)F07$$\"35L$e*[$z*RNF,$\"3$H6)e\"*)Gpt(F07$$\"3e++]iC$p
k$F,$\"3;.UkAV=_pF07$$\"3[m;H2qcZPF,$\"37PN4dUi'G'F07$$\"3O+]7.\"fF&QF
,$\"3sS)\\hOF*ecF07$$\"3Ymm;/OgbRF,$\"3wVP94I(e5&F07$$\"3w**\\ilAFjSF,
$\"3E,=I-pp%e%F07$$\"3yLLL$)*pp;%F,$\"3#fX49^(4LTF07$$\"3)RL$3xe,tUF,$
\"3#[Eb)H:C<PF07$$\"3Cn;HdO=yVF,$\"3S8ZerP;YLF07$$\"3a+++D>#[Z%F,$\"3_
0E4\"pIz.$F07$$\"3SnmT&G!e&e%F,$\"3XY(GBk>%>FF07$$\"3#RLLL)Qk%o%F,$\"3
])z7\")GQHY#F07$$\"37+]iSjE!z%F,$\"3$3T=-?kg@#F07$$\"3a+]P40O\"*[F,$\"
3h^@0%>&)H+#F07$$\"\"&F)$\"3S(=(*G_&y'z\"F0-%'COLOURG6&%$RGBG$\"#5!\"
\"F(F(-%+AXESLABELSG6$%\"tG%%x(t)G-%%VIEWG6$;F(F`_l%(DEFAULTG" 1 2 0 
1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 187 "The solution can be interprete
d as the motion of a mass in a physical system in which the mass is at
tached to a horizontal spring and is free to move on a horizontal fric
tionless surface." }}{PARA 0 "" 0 "" {TEXT -1 210 "The displacement of
 the mass first increases from the initial displacement of 1 unit and \+
then decreases to zero asymptotically. Since the displacement is alway
s positive, the spring is never under compression. " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 20 "max
imum displacement" }{TEXT -1 44 " can be calculated by determining the
 value " }{XPPEDIT 18 0 "t[max];" "6#&%\"tG6#%$maxG" }{TEXT -1 4 " of \+
" }{TEXT 290 1 "t" }{TEXT -1 11 " for which " }{XPPEDIT 18 0 "dx/dt = \+
0;" "6#/*&%#dxG\"\"\"%#dtG!\"\"\"\"!" }{TEXT -1 51 ", and then computi
ng the value of the solution at  " }{XPPEDIT 18 0 "t[max];" "6#&%\"tG6
#%$maxG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = 8/3;" "6#/%\"xG*&
\"\")\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t)-5/3;" "
6#,&-%$expG6#,$%\"tG!\"\"\"\"\"*&\"\"&F*\"\"$F)F)" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "exp(-4*t)" "6#-%$expG6#,$*&\"\"%\"\"\"%\"tGF)!\"\"" }
{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "dx/dt=-8/3" "6#/*&%
#dxG\"\"\"%#dtG!\"\",$*&\"\")F&\"\"$F(F(" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "exp(-t)+20/3" "6#,&-%$expG6#,$%\"tG!\"\"\"\"\"*&\"#?F*\"\"$F)F*
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-4*t)" "6#-%$expG6#,$*&\"\"%\"\"
\"%\"tGF)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "The de
rivative " }{XPPEDIT 18 0 "dx/dt" "6#*&%#dxG\"\"\"%#dtG!\"\"" }{TEXT 
-1 14 " is zero when " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "8/3;" "6#*&\"\")\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 
0 "exp(-t) = 20/3;" "6#/-%$expG6#,$%\"tG!\"\"*&\"#?\"\"\"\"\"$F)" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-4*t)" "6#-%$expG6#,$*&\"\"%\"\"\"%
\"tGF)!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 12 "which giv
es " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(3*t) = 20/
8;" "6#/-%$expG6#*&\"\"$\"\"\"%\"tGF)*&\"#?F)\"\")!\"\"" }{TEXT -1 2 "
. " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "t=t[max]" "6#/%\"tG&F$6#%$maxG" }{XPPEDIT 18 
0 "``=1/3" "6#/%!G*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 
0 "ln(5/2)" "6#-%#lnG6#*&\"\"&\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }
{TEXT 292 1 "~" }{TEXT -1 14 " 0.3054302440." }}{PARA 0 "" 0 "" {TEXT 
-1 42 "The corresponding maximum displacement is " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=8/3" "6#/%\"xG*&\"\")\"\"\"\"\"$!\"
\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-ln(5/2)/3)-5/3" "6#,&-%$expG6
#,$*&-%#lnG6#*&\"\"&\"\"\"\"\"#!\"\"F.\"\"$F0F0F.*&F-F.F1F0F0" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "exp(-4*ln(5/2)/3) = 8/3" "6#/-%$expG6#,$*(\"
\"%\"\"\"-%#lnG6#*&\"\"&F*\"\"#!\"\"F*\"\"$F1F1*&\"\")F*F2F1" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "(2/5)^(1/3) -5/3" "6#,&)*&\"\"#\"\"\"\"\"&!\"
\"*&F'F'\"\"$F)F'*&F(F'F+F)F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(2/5)^(
4/3) = 2*(2/5)^(1/3);" "6#/)*&\"\"#\"\"\"\"\"&!\"\"*&\"\"%F'\"\"$F)*&F
&F')*&F&F'F(F)*&F'F'F,F)F'" }{TEXT -1 1 " " }{TEXT 293 1 "~" }{TEXT 
-1 14 " 1.473612599. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 
"Solution using " }{TEXT 0 6 "dsolve" }{TEXT -1 5 " and " }{TEXT 0 7 "
desolve" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "de := diff(x(t),t$2)+5*diff(x(t),t)
+4*x(t)=0;\nic := x(0)=1,D(x)(0)=4;\ndsolve(\{de,ic\},x(t));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"
\"#\"\"\"*&\"\"&F2-F(6$F*F-F2F2*&\"\"%F2F*F2F2\"\"!" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"!\"\"\"/--%\"DG6#F(F)\"\"%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,&*&#\"\"&\"\"$\"\"\"-%$
expG6#,$*&\"\"%F-F'F-!\"\"F-F4*&#\"\")F,F--F/6#,$F'F4F-F-" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "de \+
:= diff(x(t),t$2)+5*diff(x(t),t)+4*x(t)=0;\nic := x(0)=1,D(x)(0)=4;\nd
esolve(\{de,ic\},x(t),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%#deG/,(-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&\"\"&F2-F(6$F*F-
F2F2*&\"\"%F2F*F2F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-
%\"xG6#\"\"!\"\"\"/--%\"DG6#F(F)\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6$%8auxiliary~equation~.~.~G/,(*$)%\"mG\"\"#\"\"\"F**&\"\"&F*F(F*F*\"
\"%F*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%%+roots~.~.~G!\"\"!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%6general~solution~.~.~G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,&*&&%\"CG6#\"\"\"F--%$e
xpG6#,$F'!\"\"F-F-*&&F+6#\"\"#F--F/6#,$*&\"\"%F-F'F-F2F-F-" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Afro
m~the~initial~conditions~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"
\"\",&&%\"CG6#F$F$&F'6#\"\"#F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"
\"%,&&%\"CG6#\"\"\"!\"\"*&F$F)&F'6#\"\"#F)F*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-so~that~.~.~G
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\"#\"\")\"\"$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"##!\"&\"\"$" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#
%\"tG,&*&#\"\")\"\"$\"\"\"-%$expG6#,$F'!\"\"F-F-*&#\"\"&F,F--F/6#,$*&
\"\"%F-F'F-F2F-F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 4 "The " }{TEXT 259 20 "maximum displacement" }{TEXT -1 44 " \+
can be calculated by determining the value " }{XPPEDIT 18 0 "t[max];" 
"6#&%\"tG6#%$maxG" }{TEXT -1 4 " of " }{TEXT 291 1 "t" }{TEXT -1 11 " \+
for which " }{XPPEDIT 18 0 "dx/dt = 0;" "6#/*&%#dxG\"\"\"%#dtG!\"\"\"
\"!" }{TEXT -1 51 ", and then computing the value of the solution at  \+
" }{XPPEDIT 18 0 "t[max];" "6#&%\"tG6#%$maxG" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "g
 := t -> 8/3*exp(-t)-5/3*exp(-4*t):\neq := diff(g(t),t)=0;\ntmax := so
lve(eq,t);\nevalf(evalf[14](tmax));\ng(tmax);\nsimplify(%);\nevalf(eva
lf[14](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/,&*&#\"\")\"\"$
\"\"\"-%$expG6#,$%\"tG!\"\"F+F1*&#\"#?F*F+-F-6#,$*&\"\"%F+F0F+F1F+F+\"
\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tmaxG,$*&#\"\"\"\"\"$F(-%#ln
G6##\"\"#\"\"&F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+SCIaI!#5
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"#\"\"\"\"\"&!\"\"F%#F&\"\"
$F'#F%F*F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"#\"\"\"\"\"&!\"
\"F%#F&\"\"$F'#F%F*F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*f7OZ\"!
\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 34 "Consider the
 initial value problem" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "d^2*x/(d*t^2)+7/3;" "6#,&*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F
(!\"\"F(*&\"\"(F(\"\"$F,F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+2/3*
x = 0;" "6#/,&*&%#dxG\"\"\"%#dtG!\"\"F'*(\"\"#F'\"\"$F)%\"xGF'F'\"\"!
" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "x(0) = -1;" "6#/-%\"xG6#\"\"!,$\"
\"\"!\"\"" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "Eval(dx/dt,t = 0) = 8;" "
6#/-%%EvalG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!\"\")" }{TEXT -1 2 ". \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The a
uxiliary equation is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "3*m^2+7*m+2 = 0;" "6#/,(*&\"\"$\"\"\"*$%\"mG\"\"#F'F'*&\"\"(F'F)
F'F'F*F'\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(3*m+1)*(m+2) = 0;" "
6#/*&,&*&\"\"$\"\"\"%\"mGF(F(F(F(F(,&F)F(\"\"#F(F(\"\"!" }{TEXT -1 2 "
. " }}{PARA 0 "" 0 "" {TEXT -1 14 "The solutions " }{XPPEDIT 18 0 "m =
 -1/3;" "6#/%\"mG,$*&\"\"\"F'\"\"$!\"\"F)" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "m = -2;" "6#/%\"mG,$\"\"#!\"\"" }{TEXT -1 76 " show tha
t the general solution of the homogeneous differential equation is " }
}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = C[1]*exp(-t/3
)+C[2]*exp(-2*t);" "6#/-%\"xG6#%\"tG,&*&&%\"CG6#\"\"\"F--%$expG6#,$*&F
'F-\"\"$!\"\"F4F-F-*&&F+6#\"\"#F--F/6#,$*&F8F-F'F-F4F-F-" }{TEXT -1 2 
". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "x = C[1]*exp(-t/3)+C[2]*exp(-2*t);" "6#/%\"xG
,&*&&%\"CG6#\"\"\"F*-%$expG6#,$*&%\"tGF*\"\"$!\"\"F2F*F**&&F(6#\"\"#F*
-F,6#,$*&F6F*F0F*F2F*F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 
8 "We have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt \+
= -C[1]/3;" "6#/*&%#dxG\"\"\"%#dtG!\"\",$*&&%\"CG6#F&F&\"\"$F(F(" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t/3)-2*C[2]*exp(-2*t)" "6#,&-%$exp
G6#,$*&%\"tG\"\"\"\"\"$!\"\"F,F**(\"\"#F*&%\"CG6#F.F*-F%6#,$*&F.F*F)F*
F,F*F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 22 "The initial condition " }{XPPEDIT 18 0 "x(0) = -1;" 
"6#/-%\"xG6#\"\"!,$\"\"\"!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Ev
al(dx/dt,t = 0) = 8;" "6#/-%%EvalG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!
\"\")" }{TEXT -1 29 " give rise to the equations: " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "-1 = C[1]+C[2];" "6#/,$\"\"\"!\"\",&&
%\"CG6#F%F%&F)6#\"\"#F%" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 
4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "8 = -C[1]/
3-2*C[2];" "6#/\"\"),&*&&%\"CG6#\"\"\"F*\"\"$!\"\"F,*&\"\"#F*&F(6#F.F*
F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "Substituting " }
{XPPEDIT 18 0 "C[2]=-1-C[1]" "6#/&%\"CG6#\"\"#,&\"\"\"!\"\"&F%6#F)F*" 
}{TEXT -1 55 " from the first equation in the second equation gives: \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-C[1]/3+2+2*C[1]=
8" "6#/,(*&&%\"CG6#\"\"\"F)\"\"$!\"\"F+\"\"#F)*&F,F)&F'6#F)F)F)\"\")" 
}{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "5*C[1]/3=6" "6#/*(\"\"&\"\"\"&%
\"CG6#F&F&\"\"$!\"\"\"\"'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT 
-1 8 "so that " }{XPPEDIT 18 0 "C[1]=18/5" "6#/&%\"CG6#\"\"\"*&\"#=F'
\"\"&!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "C[2]=-23/5" "6#/&%\"CG
6#\"\"#,$*&\"#B\"\"\"\"\"&!\"\"F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 20 "It follows that the " }{TEXT 259 19 "particular solution
" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x
(t) = 18/5;" "6#/-%\"xG6#%\"tG*&\"#=\"\"\"\"\"&!\"\"" }{TEXT -1 1 " " 
}{XPPEDIT 18 0 "exp(-t/3)-23/5;" "6#,&-%$expG6#,$*&%\"tG\"\"\"\"\"$!\"
\"F,F**&\"#BF*\"\"&F,F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-2*t);" "
6#-%$expG6#,$*&\"\"#\"\"\"%\"tGF)!\"\"" }{TEXT -1 1 " " }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{TEXT 284 17 "_________________" }{TEXT -1 1 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 38 "satisfies the given initial condtions
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 56 "We construct the solution and plot its gr
aph as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 92 "g := t -> 18/5*exp(-t/3)-23/5*exp(-2*t):\n'g(t
)'=g(t);\nplot(g(t),t=0..8,labels=[`t`,`x(t)`]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"gG6#%\"tG,&*&#\"#=\"\"&\"\"\"-%$expG6#,$*&\"\"$!\"
\"F'F-F4F-F-*&#\"#BF,F--F/6#,$*&\"\"#F-F'F-F4F-F4" }}{PARA 13 "" 1 "" 
{GLPLOT2D 633 305 305 {PLOTDATA 2 "6%-%'CURVESG6$7\\o7$$\"\"!F)$!2c***
************!#<7$$\"3emmm;arz@!#>$!3![cB93i$)H)!#=7$$\"39LLLL3VfVF0$!3
]z\\SRGaymF37$$\"3s******\\i9RlF0$!3]$H!>]_)p8&F37$$\"3Hmmmm;')=()F0$!
3#z*3P3XGqOF37$$\"3%*******\\#HyI\"F3$!3Kp\">ULnb[*F07$$\"3ELLLLBxV<F3
$\"3P,,*[gu5^\"F37$$\"3kmm;zS3B@F3$\"3cfO0HLUbMF37$$\"3-+++DeR-DF3$\"3
N#e\"ok#4>B&F37$$\"3SLL$3d2<)GF3$\"3Q;Xh'=?J&oF37$$\"3ymmm;$>5E$F3$\"3
\"R$)[$z,sI$)F37$$\"31+++v2<9TF3$\"3*=P?ok/%=6F,7$$\"3MLLLLAKn\\F3$\"3
')>=hWRJZ8F,7$$\"3ELLLL*Gh#eF3$\"3/!*RpEC/I:F,7$$\"3=LLLLc$\\o'F3$\"3)
ok'oB_xs;F,7$$\"34+++v0mRvF3$\"3E;7B<$*o\"y\"F,7$$\"3)emmm^&Q%R)F3$\"3
!eoUX?ZI'=F,7$$\"3w)*****\\\\#o=*F3$\"3C#)f_q5!z\">F,7$$\"3wKLL$Qk#z**
F3$\"3975;Po;c>F,7$$\"3Ymmm^*y*z5F,$\"351k[1X8\")>F,7$$\"3))*****\\YJ?
;\"F,$\"3qNC\"z*>m$*>F,7$$\"3Wmmm\"H!*oC\"F,$\"3M$z!4#*3y&*>F,7$$\"3?L
LL=\"\\<L\"F,$\"3a![4QXI)))>F,7$$\"3]mmm,eL;9F,$\"3sQN$zWEX(>F,7$$\"3
\")*****\\[A4]\"F,$\"3E=Sg\"pSU&>F,7$$\"3wmmm'3Q\\n\"F,$\"3]rdSPsT)*=F
,7$$\"3OLLLB6@G=F,$\"32=\"3w/O%Q=F,7$$\"3&)******f-w+?F,$\"33%**)yh$4P
w\"F,7$$\"3%*********y,u@F,$\"3)G)=@e7l%o\"F,7$$\"3)*******RP)4M#F,$\"
3u3%R!>572;F,7$$\"3ILLL=Zg#\\#F,$\"3.jtY*oep`\"F,7$$\"3cmmmEn*Gn#F,$\"
35M'4XO+]X\"F,7$$\"3Tmmm1xiDGF,$\"37W)>oRkuQ\"F,7$$\"3!)*****\\9!H.IF,
$\"3e.'*=coe68F,7$$\"3Immm1:bgJF,$\"335jfAA3Z7F,7$$\"3<+++X@4LLF,$\"3G
E@*o4L$z6F,7$$\"31+++N;R(\\$F,$\"3ZH(es85y6\"F,7$$\"3wmmm;4#)oOF,$\"3;
48boarc5F,7$$\"3jmmm6lCEQF,$\"3cQL5X#\\L+\"F,7$$\"3ELLL$G^g*RF,$\"3ojU
/5?W'[*F37$$\"3oKLL=2VsTF,$\"3Ca;nk%=&[*)F37$$\"3f*****\\`pfK%F,$\"36)
fqax.W])F37$$\"3!HLLLm&z\"\\%F,$\"3;e(y9o7*[!)F37$$\"3s******z-6jYF,$
\"3(=:(\\s;^.wF37$$\"3<******4#32$[F,$\"3i2bEBcK\">(F37$$\"3O*****\\#y
'G*\\F,$\"3+c)o\"fwe8oF37$$\"3G******H%=H<&F,$\"3EIg\\cT?<kF37$$\"35mm
m1>qM`F,$\"3%Qe+W,N13'F37$$\"3%)*******HSu]&F,$\"3tE\"yd(plSdF37$$\"3'
HLL$ep'Rm&F,$\"3[0))=<(z*[aF37$$\"3')******R>4NeF,$\"3#p:k`A*)p9&F37$$
\"3#emm;@2h*fF,$\"3Ygt\\O96y[F37$$\"3]*****\\c9W;'F,$\"3sLgx='Q?h%F37$
$\"3Lmmmmd'*GjF,$\"3#=4L!H`#fO%F37$$\"3j*****\\iN7]'F,$\"3Q'*yr$R;B7%F
37$$\"3aLLLt>:nmF,$\"3@Do;dLa+RF37$$\"35LLL.a#o$oF,$\"3SX+FEq2'o$F37$$
\"3ammm^Q40qF,$\"3mS*R,[I][$F37$$\"3y******z]rfrF,$\"3d*[9K3p*4LF37$$
\"3gmmmc%GpL(F,$\"3rGffK]6?JF37$$\"3/LLL8-V&\\(F,$\"3?y(Q5^]&fHF37$$\"
3=+++XhUkwF,$\"3s!G[lGXuz#F37$$\"3=+++:o<EyF,$\"3o<(H8.71l#F37$$\"\")F
)$\"39>GSn!*R,DF3-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$%\"tG
%%x(t)G-%%VIEWG6$;F(F`_l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 187 "The solution can be interpreted as the motion of a mass \+
in a physical system in which the mass is attached to a horizontal spr
ing and is free to move on a horizontal frictionless surface." }}
{PARA 0 "" 0 "" {TEXT -1 113 "The spring is initially under compressio
n with the displacement of the mass 1 unit from the equilibrium positi
on." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "The
 mass " }{TEXT 259 49 "passes through the equilibrium position just on
ce" }{TEXT -1 23 " when the displacement " }{TEXT 294 1 "x" }{TEXT -1 
27 " is zero. This occurs when " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "18/5" "6#*&\"#=\"\"\"\"\"&!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "exp(-t/3)=23/5" "6#/-%$expG6#,$*&%\"tG\"\"\"\"\"$!\"\"F
,*&\"#BF*\"\"&F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-2*t)" "6#-%$exp
G6#,$*&\"\"#\"\"\"%\"tGF)!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" 
{TEXT -1 12 "which gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "exp(5*t/3)=23/18" "6#/-%$expG6#*(\"\"&\"\"\"%\"tGF)\"\"
$!\"\"*&\"#BF)\"#=F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "H
ence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t=3/5" "6#/%
\"tG*&\"\"$\"\"\"\"\"&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(23/18)
" "6#-%#lnG6#*&\"#B\"\"\"\"#=!\"\"" }{TEXT -1 1 " " }{TEXT 295 1 "~" }
{TEXT -1 15 " 0.1470734748. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "3*(3/23)^(1/5);evalf(evalf[1
4](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"$\"\"\"\"#B!\"\"F%#
F&\"\"&F'#\"\"%F*F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'4\"='*>!\"
*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 20 "maxim
um displacement" }{TEXT -1 44 " can be calculated by determining the v
alue " }{XPPEDIT 18 0 "t[max];" "6#&%\"tG6#%$maxG" }{TEXT -1 4 " of " 
}{TEXT 285 1 "t" }{TEXT -1 11 " for which " }{XPPEDIT 18 0 "dx/dt = 0;
" "6#/*&%#dxG\"\"\"%#dtG!\"\"\"\"!" }{TEXT -1 51 ", and then computing
 the value of the solution at  " }{XPPEDIT 18 0 "t[max];" "6#&%\"tG6#%
$maxG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = 18/5;" "6#/%\"xG*&\"#=
\"\"\"\"\"&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t/3)-23/5;" "6#
,&-%$expG6#,$*&%\"tG\"\"\"\"\"$!\"\"F,F**&\"#BF*\"\"&F,F," }{TEXT -1 
1 " " }{XPPEDIT 18 0 "exp(-2*t);" "6#-%$expG6#,$*&\"\"#\"\"\"%\"tGF)!
\"\"" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "dx/dt = -6/5
;" "6#/*&%#dxG\"\"\"%#dtG!\"\",$*&\"\"'F&\"\"&F(F(" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "exp(-t)+46/5;" "6#,&-%$expG6#,$%\"tG!\"\"\"\"\"*&\"#YF*
\"\"&F)F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-2*t);" "6#-%$expG6#,$*
&\"\"#\"\"\"%\"tGF)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
15 "The derivative " }{XPPEDIT 18 0 "dx/dt" "6#*&%#dxG\"\"\"%#dtG!\"\"
" }{TEXT -1 14 " is zero when " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "6/5;" "6#*&\"\"'\"\"\"\"\"&!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "exp(-t/3) = 46/5;" "6#/-%$expG6#,$*&%\"tG\"\"\"\"\"$!\"
\"F,*&\"#YF*\"\"&F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-2*t);" "6#-%
$expG6#,$*&\"\"#\"\"\"%\"tGF)!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "
" {TEXT -1 12 "which gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "exp(5/3*t) = 46/6;" "6#/-%$expG6#*(\"\"&\"\"\"\"\"$!\"
\"%\"tGF)*&\"#YF)\"\"'F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t=t[max]
" "6#/%\"tG&F$6#%$maxG" }{XPPEDIT 18 0 "`` = 3/5;" "6#/%!G*&\"\"$\"\"
\"\"\"&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(23/3);" "6#-%#lnG6#*&
\"#B\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 287 1 "~" }{TEXT -1 13 " \+
1.222129156." }}{PARA 0 "" 0 "" {TEXT -1 42 "The corresponding maximum
 displacement is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
x = 18/5;" "6#/%\"xG*&\"#=\"\"\"\"\"&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "exp(-``(1/5)*ln(23/3))-23/5;" "6#,&-%$expG6#,$*&-%!G6#*&\"\"\"F-
\"\"&!\"\"F--%#lnG6#*&\"#BF-\"\"$F/F-F/F-*&F4F-F.F/F/" }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "exp(-``(6/5)*ln(23/3)) = 18/5;" "6#/-%$expG6#,$*&-%!
G6#*&\"\"'\"\"\"\"\"&!\"\"F.-%#lnG6#*&\"#BF.\"\"$F0F.F0*&\"#=F.F/F0" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "(3/23)^(1/5)-23/5;" "6#,&)*&\"\"$\"\"\"
\"#B!\"\"*&F'F'\"\"&F)F'*&F(F'F+F)F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "
(3/23)^(6/5) = 3*(3/23)^(1/5);" "6#/)*&\"\"$\"\"\"\"#B!\"\"*&\"\"'F'\"
\"&F)*&F&F')*&F&F'F(F)*&F'F'F,F)F'" }{TEXT -1 1 " " }{TEXT 288 1 "~" }
{TEXT -1 14 " 1.996181096. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 15 "Solution using " }{TEXT 0 6 "dsolve" }{TEXT -1 5 " and " }
{TEXT 0 7 "desolve" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "de := diff(x(t),t$2)+7/3*dif
f(x(t),t)+2/3*x(t)=0;\nic := x(0)=-1,D(x)(0)=8;\ndsolve(\{de,ic\},x(t)
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"xG6#%\"tG
-%\"$G6$F-\"\"#\"\"\"*&#\"\"(\"\"$F2-F(6$F*F-F2F2*&#F1F6F2F*F2F2\"\"!
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"!!\"\"/--%\"D
G6#F(F)\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,&*&#\"#
B\"\"&\"\"\"-%$expG6#,$*&\"\"#F-F'F-!\"\"F-F4*&#\"#=F,F--F/6#,$*&\"\"$
F4F'F-F4F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 107 "de := diff(x(t),t$2)+7/3*diff(x(t),t)+2/3*x(t)=
0;\nic := x(0)=-1,D(x)(0)=8;\ndesolve(\{de,ic\},x(t),info=true);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"xG6#%\"tG-%\"$G
6$F-\"\"#\"\"\"*&#\"\"(\"\"$F2-F(6$F*F-F2F2*&#F1F6F2F*F2F2\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"!!\"\"/--%\"DG6#F
(F)\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8auxiliary~equation~.~.~G
/,(*&\"\"$\"\"\")%\"mG\"\"#F(F(*&\"\"(F(F*F(F(F+F(\"\"!" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+roots~.
~.~G#!\"\"\"\"$!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%6general~solution~.~.~G" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%\"xG6#%\"tG,&*&&%\"CG6#\"\"\"F--%$expG6#,$*&\"\"$!
\"\"F'F-F4F-F-*&&F+6#\"\"#F--F/6#,$*&F8F-F'F-F4F-F-" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Afrom~the~ini
tial~conditions~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/!\"\",&&%\"C
G6#\"\"\"F)&F'6#\"\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"),&*&#
\"\"\"\"\"$F(&%\"CG6#F(F(!\"\"*&\"\"#F(&F+6#F/F(F-" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-so~that~.~.~
G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\"#\"#=\"\"&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"##!#B\"\"&" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#
%\"tG,&*&#\"#=\"\"&\"\"\"-%$expG6#,$*&\"\"$!\"\"F'F-F4F-F-*&#\"#BF,F--
F/6#,$*&\"\"#F-F'F-F4F-F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 25 "We can find the value of " }{TEXT 296 1 "t" }
{TEXT -1 15 " when the mass " }{TEXT 259 39 "passes through the equili
brium position" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "g := t -> 18/5*exp(-t/3)-23/
5*exp(-2*t):\ntzero := solve(g(t)=0,t);\nevalf(evalf[14](tzero));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&tzeroG,$*&#\"\"$\"\"&\"\"\"-%#lnG6#
#\"#=\"#BF*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+[Ztq9!#5" }}}
{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 20 "maximum displacement
" }{TEXT -1 44 " can be calculated by determining the value " }
{XPPEDIT 18 0 "t[max];" "6#&%\"tG6#%$maxG" }{TEXT -1 4 " of " }{TEXT 
286 1 "t" }{TEXT -1 11 " for which " }{XPPEDIT 18 0 "dx/dt = 0;" "6#/*
&%#dxG\"\"\"%#dtG!\"\"\"\"!" }{TEXT -1 51 ", and then computing the va
lue of the solution at  " }{XPPEDIT 18 0 "t[max];" "6#&%\"tG6#%$maxG" 
}{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 130 "g := t -> 18/5*exp(-t/3)-23/5*exp(-2*t):\ndiff(
g(t),t)=0;\ntmax := solve(%,t);\nevalf(evalf[14](tmax));\ng(tmax);\nev
alf(evalf[14](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&#\"\"'\"\"&
\"\"\"-%$expG6#,$*&\"\"$!\"\"%\"tGF)F0F)F0*&#\"#YF(F)-F+6#,$*&\"\"#F)F
1F)F0F)F)\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tmaxG,$*&#\"\"$\"
\"&\"\"\"-%#lnG6##F(\"#BF*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"
+c\"H@A\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"$\"\"\"\"#B!
\"\"F%#F&\"\"&F'#\"\"%F*F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'4\"
='*>!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 
"" {TEXT -1 26 "Critically damped systems " }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1 ";" }}}{PARA 258 "" 0 "" {TEXT -1 36 "Consider the di
fferential equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "d^2*x/(d*t^2)+2*lambda;" "6#,&*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"
tGF&F(!\"\"F(*&F&F(%'lambdaGF(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/d
t+omega^2*x = 0;" "6#/,&*&%#dxG\"\"\"%#dtG!\"\"F'*&%&omegaG\"\"#%\"xGF
'F'\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }
{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 5 " and " }{XPPEDIT 18 
0 "omega" "6#%&omegaG" }{TEXT -1 18 " are positive and " }{XPPEDIT 18 
0 "lambda = omega;" "6#/%'lambdaG%&omegaG" }{TEXT -1 2 ". " }}{PARA 0 
"" 0 "" {TEXT -1 27 "The auxiliary equation has " }{TEXT 259 25 "exact
ly one real solution" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "-lambda;" "6#,$
%'lambdaG!\"\"" }{TEXT -1 63 ", so the general solution of the differe
ntial equation above is" }}{PARA 256 "" 0 "" {TEXT -1 3 "   " }
{XPPEDIT 18 0 "x(t) = exp(-lambda*t)*(C[1]+C[2]*t);" "6#/-%\"xG6#%\"tG
*&-%$expG6#,$*&%'lambdaG\"\"\"F'F/!\"\"F/,&&%\"CG6#F/F/*&&F36#\"\"#F/F
'F/F/F/" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }
{XPPEDIT 18 0 "x(t)->0" "6#f*6#-%\"xG6#%\"tG7\"6$%)operatorG%&arrowG6
\"\"\"!F-F-F-" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "t->infinity" "6#f*6#
%\"tG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 20 " in t
his case also. " }}{PARA 0 "" 0 "" {TEXT -1 155 "The motion is similar
 to that of an over-damped system, but any slight decrease in the damp
ing will lead to oscillatory motion as is suggested by the term " }
{TEXT 259 17 "critically damped" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 40 "An example of a critically damped syste
m" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 34 "Consider the initial value problem" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+4;" "6#,&*(%\"dG\"\"#%\"x
G\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(\"\"%F(" }{TEXT -1 1 " " }{XPPEDIT 18 
0 "dx/dt+4*x = 0;" "6#/,&*&%#dxG\"\"\"%#dtG!\"\"F'*&\"\"%F'%\"xGF'F'\"
\"!" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "x(0) = 1" "6#/-%\"xG6#\"\"!\"\"
\"" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "Eval(dx/dt,t = 0) = 4" "6#/-%%Ev
alG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!\"\"%" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 90 "The only difference from the first exampl
e in the last section is that the coefficient of " }{XPPEDIT 18 0 "dx/
dt;" "6#*&%#dxG\"\"\"%#dtG!\"\"" }{TEXT -1 30 " has been changed from \+
5 to 4." }}{PARA 0 "" 0 "" {TEXT -1 88 "As indicated above, the soluti
on is qualitatively very similar to the previous solution." }}{PARA 0 
"" 0 "" {TEXT -1 23 "The auxiliary equation " }{XPPEDIT 18 0 "m^2+4*m+
4 = 0;" "6#/,(*$%\"mG\"\"#\"\"\"*&\"\"%F(F&F(F(F*F(\"\"!" }{TEXT -1 
28 " can be written in the form " }{XPPEDIT 18 0 "(m+2)^2=0" "6#/*$,&%
\"mG\"\"\"\"\"#F'F(\"\"!" }{TEXT -1 38 ", and so has the single real s
olution " }{XPPEDIT 18 0 "m = -2;" "6#/%\"mG,$\"\"#!\"\"" }{TEXT -1 3 
".  " }}{PARA 0 "" 0 "" {TEXT -1 52 "The general solution of the diffe
rential equation is" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 
0 "x(t) = C[1]*exp(-2*t)+C[2]*t*exp(-2*t);" "6#/-%\"xG6#%\"tG,&*&&%\"C
G6#\"\"\"F--%$expG6#,$*&\"\"#F-F'F-!\"\"F-F-*(&F+6#F3F-F'F--F/6#,$*&F3
F-F'F-F4F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }
}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = C[1]*exp(-2*t)+C
[2]*t*exp(-2*t);" "6#/%\"xG,&*&&%\"CG6#\"\"\"F*-%$expG6#,$*&\"\"#F*%\"
tGF*!\"\"F*F**(&F(6#F0F*F1F*-F,6#,$*&F0F*F1F*F2F*F*" }{TEXT -1 2 ", " 
}}{PARA 0 "" 0 "" {TEXT -1 8 "we have " }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "dx/dt = -2*C[1]*exp(-2*t)+C[2]*exp(-2*t)-2*C[2]*
t*exp(-2*t);" "6#/*&%#dxG\"\"\"%#dtG!\"\",(*(\"\"#F&&%\"CG6#F&F&-%$exp
G6#,$*&F+F&%\"tGF&F(F&F(*&&F-6#F+F&-F06#,$*&F+F&F4F&F(F&F&**F+F&&F-6#F
+F&F4F&-F06#,$*&F+F&F4F&F(F&F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The initial condition " }
{XPPEDIT 18 0 "x(0)=1" "6#/-%\"xG6#\"\"!\"\"\"" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "Eval(dx/dt,t = 0) = 4;" "6#/-%%EvalG6$*&%#dxG\"\"\"%#dt
G!\"\"/%\"tG\"\"!\"\"%" }{TEXT -1 29 " give rise to the equations: " }
}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1 = C[1];" "6#/\"\"
\"&%\"CG6#F$" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "4 = -2*C[1]+C[2];" "6
#/\"\"%,&*&\"\"#\"\"\"&%\"CG6#F(F(!\"\"&F*6#F'F(" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }{XPPEDIT 18 0 "C[1]=1" "6#/&%\"CG6
#\"\"\"F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "C[2] = 6;" "6#/&%\"CG6#
\"\"#\"\"'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 51 "Hence the \+
solution to the initial value problem is " }}{PARA 256 "" 0 "" {TEXT 
-1 2 "  " }{XPPEDIT 18 0 "x(t) = exp(-2*t)+6*t*exp(-2*t);" "6#/-%\"xG6
#%\"tG,&-%$expG6#,$*&\"\"#\"\"\"F'F/!\"\"F/*(\"\"'F/F'F/-F*6#,$*&F.F/F
'F/F0F/F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 40 "The graph of the solution is as follows." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
86 "g := t -> exp(-2*t)+6*t*exp(-2*t):\n'g(t)'=g(t);\nplot(g(t),t=0..3
,labels=[`t`,`x(t)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"
tG,&-%$expG6#,$*&\"\"#\"\"\"F'F/!\"\"F/*(\"\"'F/F'F/F)F/F/" }}{PARA 
13 "" 1 "" {GLPLOT2D 523 218 218 {PLOTDATA 2 "6%-%'CURVESG6$7gn7$$\"\"
!F)$\"\"\"F)7$$\"3')*****\\7t&pK!#>$\"3kTf8'Rd/7\"!#<7$$\"3s******\\i9
RlF/$\"3lQkEc#e;A\"F27$$\"33++vVV)RQ*F/$\"3aZs*e2sbH\"F27$$\"3/++vVA)G
A\"!#=$\"3ypfD2Fdd8F27$$\"3+++]Peui=F@$\"3q#)))GIy+f9F27$$\"37+++]$)z%
=#F@$\"3-cZ?E?#G\\\"F27$$\"3A++]i3&o]#F@$\"3]i*)3%=Sn^\"F27$$\"37+D1*y
6rm#F@$\"3TMA<`5ID:F27$$\"3.+]i:FPFGF@$\"3lr%\\gk7=`\"F27$$\"3#**\\(=U
Oj()HF@$\"3F]./HdRO:F27$$\"3%)***\\(oX*y9$F@$\"3M;Dk)[l\"R:F27$$\"35++
]ipZ'H$F@$\"3%=9)e0$4-a\"F27$$\"3#)***\\iNf]W$F@$\"3GJ%4w]s)R:F27$$\"3
_******\\<k$f$F@$\"3$*)o.n/N#Q:F27$$\"3z***\\P9CAu$F@$\"3w?BcwLPN:F27$
$\"3!)***\\(o5#*\\SF@$\"3d3xt'=me_\"F27$$\"3!)***\\P*zhdVF@$\"3Bb4OgX-
7:F27$$\"31++v$>fS*\\F@$\"3gxD#\\Ma>Z\"F27$$\"3$)***\\(=$f%GcF@$\"3F#[
4DdX+U\"F27$$\"3Q+++Dy,\"G'F@$\"3_+Wtx0yd8F27$$\"33++]7<zboF@$\"3C`%[$
[`'yH\"F27$$\"3`+++v4&G](F@$\"3!G<]Mz(*oA\"F27$$\"3!)*****\\7nD:)F@$\"
3K\"Q@)oWt`6F27$$\"3[+++D!*oy()F@$\"3!4ZUMOZG3\"F27$$\"3))***\\PpnsM*F
@$\"36INCi91>5F27$$\"3,++]siL-5F2$\"3YEh1Q->[%*F@7$$\"3-+++!R5'f5F2$\"
3tusY2jSQ))F@7$$\"3)***\\P/QBE6F2$\"3av[9I:4c\")F@7$$\"3!******\\\"o?&
=\"F2$\"3c'=w^@6$zvF@7$$\"31+]Pa&4*\\7F2$\"3c[ez\"eT!ypF@7$$\"33+]7j=_
68F2$\"3-_Z`,!etV'F@7$$\"33++vVy!eP\"F2$\"3E*f?zSuo!fF@7$$\"34+](=WU[V
\"F2$\"3K%*)H$Qz%*\\aF@7$$\"3)****\\7B>&)\\\"F2$\"3[`2>6o.*)\\F@7$$\"3
)***\\P>:mk:F2$\"3'\\]xOmmWa%F@7$$\"3'***\\iv&QAi\"F2$\"3]IxK;;&[=%F@7
$$\"31++vtLU%o\"F2$\"3OLrR)H1R#QF@7$$\"3!******\\Nm'[<F2$\"3[)\\\"=W8b
zMF@7$$\"3\"****\\(yb^6=F2$\"3)z&o2o)R#pJF@7$$\"3)***\\PMaKs=F2$\"39q)
f,P\"e#*GF@7$$\"3&****\\7TW)R>F2$\"3;ShrUQ)3h#F@7$$\"3z*****\\@80+#F2$
\"3ux=meP:zBF@7$$\"31++]7,Hl?F2$\"3w$\\l>6GD:#F@7$$\"3()**\\P4w)R7#F2$
\"3A@w\\R#RW'>F@7$$\"3;++]x%f\")=#F2$\"3!pS+f()Gix\"F@7$$\"3!)**\\P/-a
[AF2$\"3&='y)QSQXh\"F@7$$\"3/+](=Yb;J#F2$\"3_nzzB8Eg9F@7$$\"3')****\\i
@OtBF2$\"3k*yJD[fGK\"F@7$$\"3')**\\PfL'zV#F2$\"3;$e'e%\\!4#>\"F@7$$\"3
>+++!*>=+DF2$\"3)e>\\Bu_x2\"F@7$$\"3-++DE&4Qc#F2$\"3wJR2qD7;(*F/7$$\"3
=+]P%>5pi#F2$\"3g&RWB0p?w)F/7$$\"39+++bJ*[o#F2$\"3UI(fx3AY'zF/7$$\"33+
+Dr\"[8v#F2$\"3&ze6Q0*)e8(F/7$$\"3++++Ijy5GF2$\"3)\\7\\qkV^Y'F/7$$\"31
+]P/)fT(GF2$\"3jEc3Wmt;eF/7$$\"31+]i0j\"[$HF2$\"3![6?`T')\\D&F/7$$\"\"
$F)$\"30\"3mc8H'4ZF/-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$%
\"tG%%x(t)G-%%VIEWG6$;F(Fg]l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 4 "The " }{TEXT 259 20 "maximum displacement" }{TEXT -1 44 
" can be calculated by determining the value " }{XPPEDIT 18 0 "t[max];
" "6#&%\"tG6#%$maxG" }{TEXT -1 4 " of " }{TEXT 297 1 "t" }{TEXT -1 11 
" for which " }{XPPEDIT 18 0 "dx/dt = 0;" "6#/*&%#dxG\"\"\"%#dtG!\"\"
\"\"!" }{TEXT -1 51 ", and then computing the value of the solution at
  " }{XPPEDIT 18 0 "t[max];" "6#&%\"tG6#%$maxG" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 6 "Given " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "x=exp(-2*t)+6*t*exp(-2*t)" "6#/%\"xG,&-%$expG6#,$*&
\"\"#\"\"\"%\"tGF,!\"\"F,*(\"\"'F,F-F,-F'6#,$*&F+F,F-F,F.F,F," }
{XPPEDIT 18 0 "``=(1+6*t)*exp(-2*t)" "6#/%!G*&,&\"\"\"F'*&\"\"'F'%\"tG
F'F'F'-%$expG6#,$*&\"\"#F'F*F'!\"\"F'" }{TEXT -1 2 ", " }}{PARA 0 "" 
0 "" {TEXT -1 16 "it follows that " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "dx/dt = 6*exp(-2*t)-2*(1+6*t)*exp(-2*t);" "6#/*&%#dx
G\"\"\"%#dtG!\"\",&*&\"\"'F&-%$expG6#,$*&\"\"#F&%\"tGF&F(F&F&*(F1F&,&F
&F&*&F+F&F2F&F&F&-F-6#,$*&F1F&F2F&F(F&F(" }{XPPEDIT 18 0 "``=(4-12*t)*
exp(-2*t)" "6#/%!G*&,&\"\"%\"\"\"*&\"#7F(%\"tGF(!\"\"F(-%$expG6#,$*&\"
\"#F(F+F(F,F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "The der
ivative " }{XPPEDIT 18 0 "dx/dt" "6#*&%#dxG\"\"\"%#dtG!\"\"" }{TEXT 
-1 14 " is zero when " }{XPPEDIT 18 0 "4-12*t=0" "6#/,&\"\"%\"\"\"*&\"
#7F&%\"tGF&!\"\"\"\"!" }{TEXT -1 14 ", which gives " }{XPPEDIT 18 0 "t
=t[max]" "6#/%\"tG&F$6#%$maxG" }{XPPEDIT 18 0 "``=1/3" "6#/%!G*&\"\"\"
F&\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "exp(3*t) = 20/8;" "6#/-%$expG6#*&\"\"$\"\"\"%\"tGF)*&\"
#?F)\"\")!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "The co
rresponding maximum displacement is " }}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{XPPEDIT 18 0 "x=3*exp(-2/3)" "6#/%\"xG*&\"\"$\"\"\"-%$expG6#,$*&
\"\"#F'F&!\"\"F.F'" }{TEXT -1 1 " " }{TEXT 299 1 "~" }{TEXT -1 14 " 1.
540251357. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 108 "This is slightly larger than before, which is what we would ex
pect since the damping force has been reduced." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 15 "Solution using " }{TEXT 0 6 "dsolve" }
{TEXT -1 5 " and " }{TEXT 0 7 "desolve" }{TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "de := d
iff(x(t),t$2)+4*diff(x(t),t)+4*x(t)=0;\nic := x(0)=1,D(x)(0)=4;\ndsolv
e(\{de,ic\},x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diff
G6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&\"\"%F2-F(6$F*F-F2F2*&F4F2F*F2F
2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"!\"\"\"
/--%\"DG6#F(F)\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,
&-%$expG6#,$*&\"\"#\"\"\"F'F/!\"\"F/*(\"\"'F/F)F/F'F/F/" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "de :
= diff(x(t),t$2)+4*diff(x(t),t)+4*x(t)=0;\nic := x(0)=1,D(x)(0)=4;\nde
solve(\{de,ic\},x(t),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#deG/,(-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&\"\"%F2-F(6$F*F-F
2F2*&F4F2F*F2F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"x
G6#\"\"!\"\"\"/--%\"DG6#F(F)\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%
8auxiliary~equation~.~.~G/,(*$)%\"mG\"\"#\"\"\"F**&\"\"%F*F(F*F*F,F*\"
\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$%1single~root~.~.~G!\"#" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%6general~solution~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%
\"xG6#%\"tG,&*&&%\"CG6#\"\"\"F--%$expG6#,$*&\"\"#F-F'F-!\"\"F-F-*(&F+6
#F3F-F'F-F.F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%Afrom~the~initial~conditions~.~.~G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/\"\"\"&%\"CG6#F$" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/\"\"%,&*&\"\"#\"\"\"&%\"CG6#F(F(!\"\"&F*6#F'F(" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-so~that~.
~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\"F'" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"#\"\"'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,&
-%$expG6#,$*&\"\"#\"\"\"F'F/!\"\"F/*(\"\"'F/F'F/F)F/F/" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 20 "
maximum displacement" }{TEXT -1 44 " can be calculated by determining \+
the value " }{XPPEDIT 18 0 "t[max];" "6#&%\"tG6#%$maxG" }{TEXT -1 4 " \+
of " }{TEXT 298 1 "t" }{TEXT -1 11 " for which " }{XPPEDIT 18 0 "dx/dt
 = 0;" "6#/*&%#dxG\"\"\"%#dtG!\"\"\"\"!" }{TEXT -1 51 ", and then comp
uting the value of the solution at  " }{XPPEDIT 18 0 "t[max];" "6#&%\"
tG6#%$maxG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "g := t -> exp(-2*t)+6*t*exp(-2*t):
\ndiff(g(t),t)=0;\ntmax := solve(%,t);\nevalf(evalf[14](tmax));\ng(tma
x);\nevalf(evalf[14](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&\"\"
%\"\"\"-%$expG6#,$*&\"\"#F'%\"tGF'!\"\"F'F'*(\"#7F'F.F'F(F'F/\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tmaxG#\"\"\"\"\"$" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#$\"+LLLLL!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&
\"\"$\"\"\"-%$expG6##!\"#F%F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"
+d8DS:!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 
0 "" {TEXT -1 21 "Under-damped systems " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 258 "" 0 "" {TEXT -1 36 "Consider the diff
erential equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "d^2*x/(d*t^2)+2*lambda;" "6#,&*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF
&F(!\"\"F(*&F&F(%'lambdaGF(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+o
mega^2*x = 0;" "6#/,&*&%#dxG\"\"\"%#dtG!\"\"F'*&%&omegaG\"\"#%\"xGF'F'
\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }
{XPPEDIT 18 0 "0<lambda " "6#2\"\"!%'lambdaG" }{XPPEDIT 18 0 "`` < ome
ga;" "6#2%!G%&omegaG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 56 "
The solutions of the auxiliary equation are now complex:" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "m[1] = -lambda+sqrt(omega^2-lambda^2)*i;" "6#/&%\"mG6#\"\"\",&%'lam
bdaG!\"\"*&-%%sqrtG6#,&*$%&omegaG\"\"#F'*$F)F2F*F'%\"iGF'F'" }{TEXT 
-1 8 "   and  " }{XPPEDIT 18 0 "m[2] = -lambda-sqrt(omega^2-lambda^2)*
i;" "6#/&%\"mG6#\"\"#,&%'lambdaG!\"\"*&-%%sqrtG6#,&*$%&omegaGF'\"\"\"*
$F)F'F*F2%\"iGF2F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 59 "The general solution of the differential
 equation above is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "x(t) = exp(-lambda*t)*(C[1]*cos(sqrt(omega^2-lambda^2)*t)+C[2]*sin(
sqrt(omega^2-lambda^2)*t));" "6#/-%\"xG6#%\"tG*&-%$expG6#,$*&%'lambdaG
\"\"\"F'F/!\"\"F/,&*&&%\"CG6#F/F/-%$cosG6#*&-%%sqrtG6#,&*$%&omegaG\"\"
#F/*$F.F@F0F/F'F/F/F/*&&F46#F@F/-%$sinG6#*&-F;6#,&*$F?F@F/*$F.F@F0F/F'
F/F/F/F/" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 
309 33 "_________________________________" }{TEXT -1 1 " " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The motion is " }
{TEXT 259 11 "oscillatory" }{TEXT -1 28 ", but because of the factor \+
" }{XPPEDIT 18 0 "exp(-lambda*t);" "6#-%$expG6#,$*&%'lambdaG\"\"\"%\"t
GF)!\"\"" }{TEXT -1 6 ", the " }{TEXT 259 43 "amplitude of the oscilla
tions tends to zero" }{TEXT -1 4 " as " }{TEXT 272 1 "t" }{TEXT -1 19 
" tends to infinity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 
"An example of an under-damped system" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 26 "The initial value p
roblem:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t
^2)+2;" "6#,&*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(F&F(" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+26*x = 0;" "6#/,&*&%#dxG\"\"\"%#d
tG!\"\"F'*&\"#EF'%\"xGF'F'\"\"!" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "x(0
) = 3;" "6#/-%\"xG6#\"\"!\"\"$" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "Eval
(dx/dt,t = 0) = 7;" "6#/-%%EvalG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!\"
\"(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 31 "models an under-d
amped system. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 57 "The auxiliary equation for the differential equation is: \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m^2+2*m+26 = 0;" 
"6#/,(*$%\"mG\"\"#\"\"\"*&F'F(F&F(F(\"#EF(\"\"!" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 29 "Completing the square gives: " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m^2+2*m+1 = -26+1;" "6#/,(*
$%\"mG\"\"#\"\"\"*&F'F(F&F(F(F(F(,&\"#E!\"\"F(F(" }{TEXT -1 2 ", " }}
{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "(m+1)^2 = -25;" "6#/*$,&%\"mG\"\"\"F'F'\"\"#,$\"#D
!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m = -1;" "6#/%\"mG,$
\"\"\"!\"\"" }{TEXT -1 1 " " }{TEXT 300 1 "+" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "5*i;" "6#*&\"\"&\"\"\"%\"iGF%" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 53 "The general solution of the differential \+
equation is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) \+
= exp(-t)*(C[1]*cos*5*t+C[2]*sin*5*t);" "6#/-%\"xG6#%\"tG*&-%$expG6#,$
F'!\"\"\"\"\",&**&%\"CG6#F.F.%$cosGF.\"\"&F.F'F.F.**&F26#\"\"#F.%$sinG
F.F5F.F'F.F.F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = exp(-t)*(C[1]
*cos*5*t+C[2]*sin*5*t);" "6#/%\"xG*&-%$expG6#,$%\"tG!\"\"\"\"\",&**&%
\"CG6#F,F,%$cosGF,\"\"&F,F*F,F,**&F06#\"\"#F,%$sinGF,F3F,F*F,F,F," }
{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 3 "   " }{XPPEDIT 18 0 "dx/dt = -exp(-t)*(C[1]*cos*5*t+C[2]*s
in*5*t)+exp(-t)*(-5*C[1]*sin*5*t+5*C[2]*cos*5*t);" "6#/*&%#dxG\"\"\"%#
dtG!\"\",&*&-%$expG6#,$%\"tGF(F&,&**&%\"CG6#F&F&%$cosGF&\"\"&F&F/F&F&*
*&F36#\"\"#F&%$sinGF&F6F&F/F&F&F&F(*&-F,6#,$F/F(F&,&*,F6F&&F36#F&F&F;F
&F6F&F/F&F(*,F6F&&F36#F:F&F5F&F6F&F/F&F&F&F&" }{TEXT -1 1 " " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(-t)*((5*C[2]-C[1])
*cos*5*t-(C[2]+5*C[1])*sin*5*t);" "6#/%!G*&-%$expG6#,$%\"tG!\"\"\"\"\"
,&**,&*&\"\"&F,&%\"CG6#\"\"#F,F,&F36#F,F+F,%$cosGF,F1F,F*F,F,**,&&F36#
F5F,*&F1F,&F36#F,F,F,F,%$sinGF,F1F,F*F,F+F," }{TEXT -1 2 ". " }}{PARA 
0 "" 0 "" {TEXT -1 24 "The initial conditions  " }{XPPEDIT 18 0 "x(0) \+
= 3;" "6#/-%\"xG6#\"\"!\"\"$" }{TEXT -1 6 " and  " }{XPPEDIT 18 0 "Eva
l(dx/dt,t = 0) = 7;" "6#/-%%EvalG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!
\"\"(" }{TEXT -1 34 "  give rise to the two equations: " }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3 = C[1];" "6#/\"\"$&%\"CG6#\"\"
\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "7 = 5*C[2]-C[1];" "6#/\"\"(,&*&
\"\"&\"\"\"&%\"CG6#\"\"#F(F(&F*6#F(!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "
" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "C[1] = 3;" "6#/&%\"CG6#\"\"
\"\"\"$" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "C[2] = 2;" "6#/&%\"CG6#\"
\"#F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "The particular \+
solution " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = e
xp(-t)*(3*cos*5*t+2*sin*5*t);" "6#/-%\"xG6#%\"tG*&-%$expG6#,$F'!\"\"\"
\"\",&**\"\"$F.%$cosGF.\"\"&F.F'F.F.**\"\"#F.%$sinGF.F3F.F'F.F.F." }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 44 "satisfies the requisite \+
initial conditions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 40 "The graph of the solution is as follows." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "g := t
 -> 3*exp(-t)*cos(5*t)+2*exp(-t)*sin(5*t):\n'g(t)'=g(t);\nplot(g(t),t=
0..4,labels=[`t`,`x(t)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6
#%\"tG,&*(\"\"#\"\"\"-%$expG6#,$F'!\"\"F+-%$sinG6#,$*&\"\"&F+F'F+F+F+F
+*(\"\"$F+F,F+-%$cosGF3F+F+" }}{PARA 13 "" 1 "" {GLPLOT2D 542 306 306 
{PLOTDATA 2 "6%-%'CURVESG6$7\\q7$$\"\"!F)$\"\"$F)7$$\"3emmm;arz@!#>$\"
3wusV`/vIJ!#<7$$\"39LLLL3VfVF/$\"3Jb?N$**=\"=KF27$$\"3WmmmT&)G\\aF/$\"
39MfE.XwXKF27$$\"3s******\\i9RlF/$\"39\"yLD?,HE$F27$$\"3+LLLeR+HwF/$\"
3e)=?&QLqpKF27$$\"3Hmmmm;')=()F/$\"3OvG*Q'HPmKF27$$\"3-++]7z>^7!#=$\"3
!pg*)3Xe#zJF27$$\"3RLLLe'40j\"FO$\"3IL1olQL%)HF27$$\"3/++](Q&3d?FO$\"3
O\\ROkj#\\l#F27$$\"3mmmm;6m$[#FO$\"3EUlnt9`KAF27$$\"3Ommm\"zi$)p#FO$\"
3;[_Cg68$*>F27$$\"3jmmmmW18HFO$\"327Q#Q8M/u\"F27$$\"3*omm;9mx7$FO$\"31
(ycy#3'zZ\"F27$$\"3fmmm;yYULFO$\"3RPZo#G)>47F27$$\"3KLL3_!\\hb$FO$\"31
2&RByK&)Q*FO7$$\"3/++](GI)pPFO$\"35sM8.R=*o'FO7$$\"3Amm\"H_6N)RFO$\"3(
4_gG&Q5DSFO7$$\"3%HLL$eF>(>%FO$\"3$>LV')*ouD9FO7$$\"3Q*****\\Z7Mf%FO$!
3oI=&e?l58$FO7$$\"3Qmmm\">K'*)\\FO$!3/\")*3YUi#=sFO7$$\"3IKLLeZ*)*R&FO
$!3IOxL_iM#3\"F27$$\"3P*****\\Kd,\"eFO$!31P#3*))o:p8F27$$\"39KLLe9XMiF
O$!3]`bH+Ao\"e\"F27$$\"3-mmm\"fX(emFO$!3&)\\`&4B9iq\"F27$$\"35KL$e%*yW
w'FO$!3s!)o<Q;lB<F27$$\"3E*******H7-(oFO$!3FQr;rywN<F27$$\"3Ymm;ac%f(p
FO$!3:@**zL%REu\"F27$$\"3_KLL3!z;3(FO$!3k)e1+JcVu\"F27$$\"3g)***\\iBT(
=(FO$!35>['HU@5u\"F27$$\"3ylmm;d9$H(FO$!3T&G(Ro+vK<F27$$\"3'HLL32z))R(
FO$!3I:::C$p'><F27$$\"3.*****\\U7Y](FO$!3Kx\\DQu\">q\"F27$$\"3Wmm;H9lR
zFO$!3E;q<()*zIe\"F27$$\"3'QLLLV!pu$)FO$!3UhspgU%)*R\"F27$$\"3K+++DI(y
v)FO$!3+H9z)f%='>\"F27$$\"3xmmm;c0T\"*FO$!3M[Y.1FIP'*FO7$$\"3+LLLe%GCd
*FO$!37MAyR.]3oFO7$$\"3#*******H,Q+5F2$!3-K!oj!\\>**QFO7$$\"3&*******R
XpV5F2$!3'34>))fg*Q5FO7$$\"3)*******\\*3q3\"F2$\"3!*Q=/**RFK;FO7$$\"3(
)******4/vG6F2$\"3/sQX)QZf#RFO7$$\"3)*******p=\\q6F2$\"3'\\$\\:.z#f'eF
O7$$\"3@LLe9rR37F2$\"3iq8g!G3SF(FO7$$\"3mmm;fBIY7F2$\"3hwugh0T=$)FO7$$
\"3OL$3-')Q)o7F2$\"3Wzc5=Z*=w)FO7$$\"33++Dh`P\"H\"F2$\"3-'R!R8Q#G2*FO7
$$\"3KL3x6Ok-8F2$\"3K9.HL')3z\"*FO7$$\"3dm;Hi=\"RJ\"F2$\"3KIHkM&HID*FO
7$$\"3\")*\\7G6!=D8F2$\"31Io#G`D^H*FO7$$\"3GLLLj$[kL\"F2$\"3&\\s`T*)[f
I*FO7$$\"3?LL$etRbN\"F2$\"3y9_![66[D*FO7$$\"3MLLL36ju8F2$\"3IPw%4:2'>
\"*FO7$$\"3ELL$3[APR\"F2$\"3pFqE'[UZ!*)FO7$$\"3?LLL`Q\"GT\"F2$\"3tQG$3
l(>:')FO7$$\"3mmm\"HYHsX\"F2$\"3yo!)z5'HSo(FO7$$\"3!*****\\s]k,:F2$\"3
%R'3p<M\\ckFO7$$\"39LLL`dF!e\"F2$\"3gHNw%e_5#QFO7$$\"33++]sgam;F2$\"3l
YkuC!*HEuF/7$$\"3/++]<ep[<F2$!3i+fFl!eH'=FO7$$\"3QLLLe/TM=F2$!3g.()***
Qa.%QFO7$$\"3YLL3dowt=F2$!3r,YS9DTFWFO7$$\"3JLL$eDBJ\">F2$!3U'\\carV\"
*z%FO7$$\"3`m;H_)[V$>F2$!3(otab=#f5\\FO7$$\"3)****\\([Wdb>F2$!33n>1Yi9
h\\FO7$$\"3UL$3_/+o(>F2$!3m=Do@wv_\\FO7$$\"3immmTc-)*>F2$!3_nw>WA(z)[F
O7$$\"3[mmT+07U?F2$!3')\\&y()[zqe%FO7$$\"3Mmm;f`@'3#F2$!3cYX%Qhe04%FO7
$$\"3y****\\nZ)H;#F2$!3OhJ#G1m>)GFO7$$\"3YmmmJy*eC#F2$!3=uiRNh3B8FO7$$
\"3')******R^bJBF2$\"3-+Mis5]'f#F/7$$\"3f*****\\5a`T#F2$\"3#oxpwcv\\_
\"FO7$$\"3k***\\(3S*eX#F2$\"3QyWAh7)G)>FO7$$\"3o****\\7RV'\\#F2$\"3\"=
St*\\*GSK#FO7$$\"3m***\\PcY9a#F2$\"3aAKU:JIgDFO7$$\"3k*****\\@fke#F2$
\"3!Rh9WL=yk#FO7$$\"3Nmm;%30pi#F2$\"33_Y>n!ojg#FO7$$\"3/LLL`4NnEF2$\"3
=sS3H_-iCFO7$$\"3Emmm^b`5FF2$\"3Yw([WQR+@#FO7$$\"3#*******\\,s`FF2$\"3
klQs4dlw=FO7$$\"3[mm;zM)>$GF2$\"3\"[y^av>u8\"FO7$$\"3$*******pfa<HF2$
\"3)[/+LV,3n#F/7$$\"3#HLLeg`!)*HF2$!37&e?^=)R7ZF/7$$\"3w****\\#G2A3$F2
$!3sQ$*=@2>Z5FO7$$\"3;LLL$)G[kJF2$!3S\"znFyd_N\"FO7$$\"3#)****\\7yh]KF
2$!39'zJuSweR\"FO7$$\"3xmmm')fdLLF2$!37B8Xat,)>\"FO7$$\"3bmmm,FT=MF2$!
3=\"=u6QjME)F/7$$\"3FLL$e#pa-NF2$!3#Gu#eR7(\\w$F/7$$\"3!*******Rv&)zNF
2$\"3oNrBN$Q*oK!#?7$$\"3ILLLGUYoOF2$\"3<c1k7)*)>@%F/7$$\"3_mmm1^rZPF2$
\"3Bnqu@n$e]'F/7$$\"34++]sI@KQF2$\"3Pw_z<tc8vF/7$$\"34++]2%)38RF2$\"3a
6T;f5gWrF/7$$\"\"%F)$\"3ezc%=@?le&F/-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%
+AXESLABELSG6$%\"tG%%x(t)G-%%VIEWG6$;F(Fail%(DEFAULTG" 1 2 0 1 10 0 2 
9 1 4 2 1.000000 46.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 187 "The solution can be interpreted as the motion \+
of a mass in a physical system in which the mass is attached to a hori
zontal spring and is free to move on a horizontal frictionless surface
." }}{PARA 0 "" 0 "" {TEXT -1 100 "The mass oscillates back and forth \+
about the equilibrium position, but with a decreasing amplitude. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }
{TEXT 259 20 "maximum displacement" }{TEXT -1 44 " can be calculated b
y determining the value " }{XPPEDIT 18 0 "t[max];" "6#&%\"tG6#%$maxG" 
}{TEXT -1 4 " of " }{TEXT 302 1 "t" }{TEXT -1 11 " for which " }
{XPPEDIT 18 0 "dx/dt = 0;" "6#/*&%#dxG\"\"\"%#dtG!\"\"\"\"!" }{TEXT 
-1 51 ", and then computing the value of the solution at  " }{XPPEDIT 
18 0 "t[max];" "6#&%\"tG6#%$maxG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 6 "Given " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 
"x = exp(-t)*(3*cos*5*t+2*sin*5*t);" "6#/%\"xG*&-%$expG6#,$%\"tG!\"\"
\"\"\",&**\"\"$F,%$cosGF,\"\"&F,F*F,F,**\"\"#F,%$sinGF,F1F,F*F,F,F," }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 16 "it follows that " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt=-exp(-t)*(3*cos
*5*t+2*sin*5*t)+exp(-t)*(-15*sin*5*t+10*cos*5*t)" "6#/*&%#dxG\"\"\"%#d
tG!\"\",&*&-%$expG6#,$%\"tGF(F&,&**\"\"$F&%$cosGF&\"\"&F&F/F&F&**\"\"#
F&%$sinGF&F4F&F/F&F&F&F(*&-F,6#,$F/F(F&,&**\"#:F&F7F&F4F&F/F&F(**\"#5F
&F3F&F4F&F/F&F&F&F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "``=exp(-t)*(7*cos*5*t-17*sin*5*t)" "6#/%!G*&-%$expG6
#,$%\"tG!\"\"\"\"\",&**\"\"(F,%$cosGF,\"\"&F,F*F,F,**\"#<F,%$sinGF,F1F
,F*F,F+F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 15 "The derivative " }{XPPEDIT 18 0 "dx/dt" "6#*&%#dx
G\"\"\"%#dtG!\"\"" }{TEXT -1 14 " is zero when " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "7*cos*5*t=17*sin*5*t" "6#/**\"\"(\"\"\"
%$cosGF&\"\"&F&%\"tGF&**\"#<F&%$sinGF&F(F&F)F&" }{TEXT -1 2 ", " }}
{PARA 0 "" 0 "" {TEXT -1 14 "that is, when " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "tan*5*t=7/17" "6#/*(%$tanG\"\"\"\"\"&F&
%\"tGF&*&\"\"(F&\"#<!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 60 "The smallest positive solution of this equation is given by " }
}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "t=t[max]" "6#/%\"tG
&F$6#%$maxG" }{XPPEDIT 18 0 "`` = 1/5;" "6#/%!G*&\"\"\"F&\"\"&!\"\"" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(7/17)" "6#-%'arctanG6#*&\"\"(\"
\"\"\"#<!\"\"" }{TEXT -1 1 " " }{TEXT 305 1 "~" }{TEXT -1 16 " 0.07812
140874. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
5 "When " }{XPPEDIT 18 0 "t=t[max]" "6#/%\"tG&F$6#%$maxG" }{TEXT -1 2 
", " }{XPPEDIT 18 0 "cos*5*t=17/(13*sqrt(2))" "6#/*(%$cosG\"\"\"\"\"&F
&%\"tGF&*&\"#<F&*&\"#8F&-%%sqrtG6#\"\"#F&!\"\"" }{TEXT -1 7 "  and  " 
}{XPPEDIT 18 0 "sin*5*t=7/(13*sqrt(2))" "6#/*(%$sinG\"\"\"\"\"&F&%\"tG
F&*&\"\"(F&*&\"#8F&-%%sqrtG6#\"\"#F&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The corresponding
 maximum displacement is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "x=exp(-t[max])" "6#/%\"xG-%$expG6#,$&%\"tG6#%$maxG!\"\"
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(51+14)/(13*sqrt(2)) = 5/sqrt(2);
" "6#/*&-%!G6#,&\"#^\"\"\"\"#9F*F**&\"#8F*-%%sqrtG6#\"\"#F*!\"\"*&\"\"
&F*-F/6#F1F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-arctan(7/17)/5)" "6
#-%$expG6#,$*&-%'arctanG6#*&\"\"(\"\"\"\"#<!\"\"F-\"\"&F/F/" }{TEXT 
-1 1 " " }{TEXT 304 1 "~" }{TEXT -1 14 " 3.269846080. " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "The largest displac
ement in the negative direction is given by second positive solution o
f the equation  " }{XPPEDIT 18 0 "tan*5*t" "6#*(%$tanG\"\"\"\"\"&F%%\"
tGF%" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "7/17" "6#*&\"\"(\"\"\"\"#<!\"
\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 "This solution is " 
}{XPPEDIT 18 0 "t[min]=t[max]+Pi/5" "6#/&%\"tG6#%$minG,&&F%6#%$maxG\"
\"\"*&%#PiGF,\"\"&!\"\"F," }{TEXT -1 1 " " }{TEXT 306 1 "~" }{TEXT -1 
15 " 0.7064399395. " }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 
18 0 "t = t[min];" "6#/%\"tG&F$6#%$minG" }{TEXT -1 2 ", " }{XPPEDIT 
18 0 "cos*5*t = -17/(13*sqrt(2));" "6#/*(%$cosG\"\"\"\"\"&F&%\"tGF&,$*
&\"#<F&*&\"#8F&-%%sqrtG6#\"\"#F&!\"\"F2" }{TEXT -1 7 "  and  " }
{XPPEDIT 18 0 "sin*5*t = -7/(13*sqrt(2));" "6#/*(%$sinG\"\"\"\"\"&F&%
\"tGF&,$*&\"\"(F&*&\"#8F&-%%sqrtG6#\"\"#F&!\"\"F2" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "The corre
sponding displacement is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "x = -exp(-t[min]);" "6#/%\"xG,$-%$expG6#,$&%\"tG6#%$min
G!\"\"F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(51+14)/(13*sqrt(2)) = -5/
sqrt(2);" "6#/*&-%!G6#,&\"#^\"\"\"\"#9F*F**&\"#8F*-%%sqrtG6#\"\"#F*!\"
\",$*&\"\"&F*-F/6#F1F2F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-(arctan
(7/17)+Pi)/5);" "6#-%$expG6#,$*&,&-%'arctanG6#*&\"\"(\"\"\"\"#<!\"\"F.
%#PiGF.F.\"\"&F0F0" }{TEXT -1 1 " " }{TEXT 307 1 "~" }{TEXT -1 2 "  " 
}{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 12 ".744423943. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "Solution using " }{TEXT 
0 6 "dsolve" }{TEXT -1 5 " and " }{TEXT 0 7 "desolve" }{TEXT -1 1 " " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 92 "de := diff(x(t),t$2)+2*diff(x(t),t)+26*x(t)=0;\nic := x(0)=3,D(x
)(0)=7;\ndsolve(\{de,ic\},x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#deG/,(-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2
*&\"#EF2F*F2F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG
6#\"\"!\"\"$/--%\"DG6#F(F)\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%
\"xG6#%\"tG,&*(\"\"#\"\"\"-%$expG6#,$F'!\"\"F+-%$sinG6#,$*&\"\"&F+F'F+
F+F+F+*(\"\"$F+F,F+-%$cosGF3F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "de := diff(x(t),t$2)+2/3*di
ff(x(t),t)+37/9*x(t)=0;\nic := x(0)=3,D(x)(0)=7;\ndesolve(\{de,ic\},x(
t),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-
%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&#F1\"\"$F2-F(6$F*F-F2F2*&#\"#P\"\"*
F2F*F2F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"
!\"\"$/--%\"DG6#F(F)\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8auxilia
ry~equation~.~.~G/,(*&\"\"*\"\"\")%\"mG\"\"#F(F(*&\"\"'F(F*F(F(\"#PF(
\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%%+roots~.~.~G^$#!\"\"\"\"$\"\"#^$F%!\"#" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%6general~s
olution~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,&*(&%
\"CG6#\"\"\"F--%$expG6#,$*&\"\"$!\"\"F'F-F4F--%$sinG6#,$*&\"\"#F-F'F-F
-F-F-*(&F+6#F:F-F.F--%$cosGF7F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%
!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Afrom~the~initial~conditions~.~
.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"$&%\"CG6#\"\"#" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/\"\"(,&*&\"\"#\"\"\"&%\"CG6#F(F(F(*&#F(\"\"
$F(&F*6#F'F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%-so~that~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/&%\"CG6#\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"
\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"xG6#%\"tG,&*(\"\"%\"\"\"-%$expG6#,$*&\"\"$!\"\"F'F
+F2F+-%$sinG6#,$*&\"\"#F+F'F+F+F+F+*(F1F+F,F+-%$cosGF5F+F+" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 
259 20 "maximum displacement" }{TEXT -1 44 " can be calculated by dete
rmining the value " }{XPPEDIT 18 0 "t[max];" "6#&%\"tG6#%$maxG" }
{TEXT -1 4 " of " }{TEXT 301 1 "t" }{TEXT -1 11 " for which " }
{XPPEDIT 18 0 "dx/dt = 0;" "6#/*&%#dxG\"\"\"%#dtG!\"\"\"\"!" }{TEXT 
-1 51 ", and then computing the value of the solution at  " }{XPPEDIT 
18 0 "t[max];" "6#&%\"tG6#%$maxG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "g := t -> 3
*exp(-t)*cos(5*t)+2*exp(-t)*sin(5*t):\ndiff(g(t),t)=0;\ntmax := solve(
%,t);\nevalf(evalf[14](tmax));\ng(tmax);\nevalf(evalf[14](%));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*(\"#<\"\"\"-%$expG6#,$%\"tG!\"\"F'
-%$sinG6#,$*&\"\"&F'F,F'F'F'F-*(\"\"(F'F(F'-%$cosGF0F'F'\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tmaxG,$*&#\"\"\"\"\"&F(-%'arctanG6#
#\"\"(\"#<F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+u397y!#6" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"&\"\"#\"\"\"*&-%$expG6#,$*&#F
(F&F(-%'arctanG6##\"\"(\"#<F(!\"\"F(F'#F(F'F(F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+!3Y)pK!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 46 "This time there are infinitely many value
s of " }{TEXT 303 1 "t" }{TEXT -1 11 " for which " }{XPPEDIT 18 0 "dx/
dt = 0;" "6#/*&%#dxG\"\"\"%#dtG!\"\"\"\"!" }{TEXT -1 67 ", but it is c
lear that this calculation has obtained the first one." }}{PARA 0 "" 
0 "" {TEXT -1 75 "The maximum displacement in the negative direction c
an be calculated using " }{TEXT 0 6 "fsolve" }{TEXT -1 2 ". " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "g
 := t -> 3*exp(-t)*cos(5*t)+2*exp(-t)*sin(5*t):\ndiff(g(t),t)=0;\ntmin
 := evalf[14](fsolve(%,t=.5..1));\nevalf(evalf[14](g(tmin)));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*(\"#<\"\"\"-%$expG6#,$%\"tG!\"\"F'
-%$sinG6#,$*&\"\"&F'F,F'F'F'F-*(\"\"(F'F(F'-%$cosGF0F'F'\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tminG$\"/]d%R*Rkq!#9" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$!+VRUW<!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 85 "The q
uasi period and quasi angular frequency of the motion of an under-damp
ed system " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "
" 0 "" {TEXT -1 13 "The solution " }}{PARA 256 "" 0 "" {TEXT -1 1 " " 
}{XPPEDIT 18 0 "x(t) = exp(-lambda*t)*(C[1]*cos(sqrt(omega^2-lambda^2)
*t)+C[2]*sin(sqrt(omega^2-lambda^2)*t))" "6#/-%\"xG6#%\"tG*&-%$expG6#,
$*&%'lambdaG\"\"\"F'F/!\"\"F/,&*&&%\"CG6#F/F/-%$cosG6#*&-%%sqrtG6#,&*$
%&omegaG\"\"#F/*$F.F@F0F/F'F/F/F/*&&F46#F@F/-%$sinG6#*&-F;6#,&*$F?F@F/
*$F.F@F0F/F'F/F/F/F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "c
an be written in the alternative form " }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "x(t) = A*exp(-lambda*t)*sin(sqrt(omega^2-lambda^
2)*t+phi);" "6#/-%\"xG6#%\"tG*(%\"AG\"\"\"-%$expG6#,$*&%'lambdaGF*F'F*
!\"\"F*-%$sinG6#,&*&-%%sqrtG6#,&*$%&omegaG\"\"#F**$F0F=F1F*F'F*F*%$phi
GF*F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "A=sqrt(C[1]^2+C[2]^2)" "6#/%
\"AG-%%sqrtG6#,&*$&%\"CG6#\"\"\"\"\"#F-*$&F+6#F.F.F-" }{TEXT -1 22 ", \+
and the phase angle " }{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 33 " i
s determined by the equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "sin*phi=C[1]/A" "6#/*&%$sinG\"\"\"%$phiGF&*&&%\"CG6#F&F
&%\"AG!\"\"" }{TEXT -1 4 ",   " }{XPPEDIT 18 0 "cos*phi=C[2]/A" "6#/*&
%$cosG\"\"\"%$phiGF&*&&%\"CG6#\"\"#F&%\"AG!\"\"" }{TEXT -1 8 "  and   \+
" }{XPPEDIT 18 0 "tan*phi = C[1]/C[2]" "6#/*&%$tanG\"\"\"%$phiGF&*&&%
\"CG6#F&F&&F*6#\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 74 "The solution can be thought of as a sine wave with \"decreasing
 amplitude\" " }{XPPEDIT 18 0 "A*exp(-lambda*t)" "6#*&%\"AG\"\"\"-%$ex
pG6#,$*&%'lambdaGF%%\"tGF%!\"\"F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 4 "Let " }{XPPEDIT 18 0 "omega[0]=sqrt(omega^2-lambda^2)" "6#
/&%&omegaG6#\"\"!-%%sqrtG6#,&*$F%\"\"#\"\"\"*$%'lambdaGF-!\"\"" }
{TEXT -1 9 " so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "x(t)=A*exp(-lambda*t)*sin(omega[0]*t+phi)" "6#/-%\"xG6#%\"tG*(%
\"AG\"\"\"-%$expG6#,$*&%'lambdaGF*F'F*!\"\"F*-%$sinG6#,&*&&%&omegaG6#
\"\"!F*F'F*F*%$phiGF*F*" }{TEXT -1 1 "." }}{PARA 258 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "omega[0]" "6#&%&omegaG6#\"\"!" }{TEXT -1 8 " is \+
the " }{TEXT 259 23 "quasi angular frequency" }{TEXT -1 37 " of the so
lution, with corresponding " }{TEXT 259 12 "quasi period" }{TEXT -1 1 
" " }{XPPEDIT 18 0 "2*Pi/omega[0]" "6#*(\"\"#\"\"\"%#PiGF%&%&omegaG6#
\"\"!!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 55 "Note that \+
the quasi angular frequency is less than the " }{TEXT 259 25 "natural \+
angular frequency" }{TEXT -1 1 " " }{XPPEDIT 18 0 "omega" "6#%&omegaG
" }{TEXT -1 37 " of the system with damping removed. " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "To justify this termin
ology we note that, even though " }{XPPEDIT 18 0 "x(t)" "6#-%\"xG6#%\"
tG" }{TEXT -1 57 " is not periodic, it has the same zeros as the sine \+
wave " }{XPPEDIT 18 0 "sin(omega[0]*t+phi);" "6#-%$sinG6#,&*&&%&omegaG
6#\"\"!\"\"\"%\"tGF,F,%$phiGF," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 17 "They occur where " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "omega[0]*t+phi=k*Pi" "6#/,&*&&%&omegaG6#\"\"!\"\"\"%\"t
GF*F*%$phiGF**&%\"kGF*%#PiGF*" }{TEXT -1 4 ",   " }{XPPEDIT 18 0 "k=` \+
. . . `,-2,-1,0,1,2,` . . . `" "6)/%\"kG%(~.~.~.~G,$\"\"#!\"\",$\"\"\"
F(\"\"!F*F'F%" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 15 "that is
, where " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t=(k*Pi-p
hi)/omega[0]" "6#/%\"tG*&,&*&%\"kG\"\"\"%#PiGF)F)%$phiG!\"\"F)&%&omega
G6#\"\"!F," }{TEXT -1 4 ",   " }{XPPEDIT 18 0 "k=` . . . `,-2,-1,0,1,2
,` . . . `" "6)/%\"kG%(~.~.~.~G,$\"\"#!\"\",$\"\"\"F(\"\"!F*F'F%" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 "Hence the spacing betwe
en successive zeros is " }{XPPEDIT 18 0 "Pi/omega[0]" "6#*&%#PiG\"\"\"
&%&omegaG6#\"\"!!\"\"" }{TEXT -1 38 ", or half the period of the sine \+
wave " }{XPPEDIT 18 0 "sin(omega[0]*t+phi);" "6#-%$sinG6#,&*&&%&omegaG
6#\"\"!\"\"\"%\"tGF,F,%$phiGF," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 6 "Since " }{XPPEDIT 18 0 "-1<=sin(omega[0]*t+phi)" "6#1,$\"
\"\"!\"\"-%$sinG6#,&*&&%&omegaG6#\"\"!F%%\"tGF%F%%$phiGF%" }{XPPEDIT 
18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 9 " for all " }{TEXT 310 1 "t" }
{TEXT -1 15 ", the graph of " }{XPPEDIT 18 0 "x=A*exp(-lambda*t)*sin(o
mega[0]*t+phi)" "6#/%\"xG*(%\"AG\"\"\"-%$expG6#,$*&%'lambdaGF'%\"tGF'!
\"\"F'-%$sinG6#,&*&&%&omegaG6#\"\"!F'F.F'F'%$phiGF'F'" }{TEXT -1 38 " \+
lies entirely between the two curves " }{XPPEDIT 18 0 "x=A*exp(-lambda
*t)" "6#/%\"xG*&%\"AG\"\"\"-%$expG6#,$*&%'lambdaGF'%\"tGF'!\"\"F'" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "x=-A*exp(-lambda*t)" "6#/%\"xG,$*&%
\"AG\"\"\"-%$expG6#,$*&%'lambdaGF(%\"tGF(!\"\"F(F0" }{TEXT -1 2 ". " }
}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "x = A*exp(-lambda
*t)*sin(omega[0]*t+phi)" "6#/%\"xG*(%\"AG\"\"\"-%$expG6#,$*&%'lambdaGF
'%\"tGF'!\"\"F'-%$sinG6#,&*&&%&omegaG6#\"\"!F'F.F'F'%$phiGF'F'" }
{TEXT -1 3 ",  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx
/dt = -A*lambda*exp(-lambda*t)*sin(omega[0]*t+phi)+A*omega[0]*exp(-lam
bda*t)*cos(omega[0]*t+phi);" "6#/*&%#dxG\"\"\"%#dtG!\"\",&**%\"AGF&%'l
ambdaGF&-%$expG6#,$*&F,F&%\"tGF&F(F&-%$sinG6#,&*&&%&omegaG6#\"\"!F&F2F
&F&%$phiGF&F&F(**F+F&&F96#F;F&-F.6#,$*&F,F&F2F&F(F&-%$cosG6#,&*&&F96#F
;F&F2F&F&F<F&F&F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " 
}{XPPEDIT 18 0 "``=A*exp(-lambda*t)*(omega[0]*cos(omega[0]*t+phi)-lamb
da*sin(omega[0]*t+phi))" "6#/%!G*(%\"AG\"\"\"-%$expG6#,$*&%'lambdaGF'%
\"tGF'!\"\"F',&*&&%&omegaG6#\"\"!F'-%$cosG6#,&*&&F36#F5F'F.F'F'%$phiGF
'F'F'*&F-F'-%$sinG6#,&*&&F36#F5F'F.F'F'F=F'F'F/F'" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that
 " }{XPPEDIT 18 0 "sin(omega[0]*t+phi)=1" "6#/-%$sinG6#,&*&&%&omegaG6#
\"\"!\"\"\"%\"tGF-F-%$phiGF-F-" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "o
mega[0]*t+phi = Pi/2+k*Pi;" "6#/,&*&&%&omegaG6#\"\"!\"\"\"%\"tGF*F*%$p
hiGF*,&*&%#PiGF*\"\"#!\"\"F**&%\"kGF*F/F*F*" }{TEXT -1 8 ", where " }
{XPPEDIT 18 0 "k=` . . . `,-2,-1,0,1,2,` . . . `" "6)/%\"kG%(~.~.~.~G,
$\"\"#!\"\",$\"\"\"F(\"\"!F*F'F%" }{TEXT -1 10 ", so that " }{XPPEDIT 
18 0 "cos(omega[0]*t+phi)=0" "6#/-%$cosG6#,&*&&%&omegaG6#\"\"!\"\"\"%
\"tGF-F-%$phiGF-F," }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 29 "T
his happens where the curve " }{XPPEDIT 18 0 "x = A*exp(-lambda*t)*sin
(omega[0]*t+phi)" "6#/%\"xG*(%\"AG\"\"\"-%$expG6#,$*&%'lambdaGF'%\"tGF
'!\"\"F'-%$sinG6#,&*&&%&omegaG6#\"\"!F'F.F'F'%$phiGF'F'" }{TEXT -1 17 
" meets the curve " }{XPPEDIT 18 0 "x = A*exp(-lambda*t)" "6#/%\"xG*&%
\"AG\"\"\"-%$expG6#,$*&%'lambdaGF'%\"tGF'!\"\"F'" }{TEXT -1 53 ", and \+
the gradient of both curves at these points is " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt = -A*lambda*exp(-lambda*t);" "6#/
*&%#dxG\"\"\"%#dtG!\"\",$*(%\"AGF&%'lambdaGF&-%$expG6#,$*&F,F&%\"tGF&F
(F&F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Similarly " }
{XPPEDIT 18 0 "sin(omega[0]*t+phi) = -1;" "6#/-%$sinG6#,&*&&%&omegaG6#
\"\"!\"\"\"%\"tGF-F-%$phiGF-,$F-!\"\"" }{TEXT -1 6 " when " }{XPPEDIT 
18 0 "omega[0]*t+phi = -Pi/2+k*Pi;" "6#/,&*&&%&omegaG6#\"\"!\"\"\"%\"t
GF*F*%$phiGF*,&*&%#PiGF*\"\"#!\"\"F1*&%\"kGF*F/F*F*" }{TEXT -1 8 ", wh
ere " }{XPPEDIT 18 0 "k=` . . . `,-2,-1,0,1,2,` . . . `" "6)/%\"kG%(~.
~.~.~G,$\"\"#!\"\",$\"\"\"F(\"\"!F*F'F%" }{TEXT -1 10 ", so that " }
{XPPEDIT 18 0 "cos(omega[0]*t+phi)=0" "6#/-%$cosG6#,&*&&%&omegaG6#\"\"
!\"\"\"%\"tGF-F-%$phiGF-F," }{TEXT -1 7 " also. " }}{PARA 0 "" 0 "" 
{TEXT -1 29 "This happens where the curve " }{XPPEDIT 18 0 "x = A*exp(
-lambda*t)*sin(omega[0]*t+phi)" "6#/%\"xG*(%\"AG\"\"\"-%$expG6#,$*&%'l
ambdaGF'%\"tGF'!\"\"F'-%$sinG6#,&*&&%&omegaG6#\"\"!F'F.F'F'%$phiGF'F'
" }{TEXT -1 17 " meets the curve " }{XPPEDIT 18 0 "x = -A*exp(-lambda*
t);" "6#/%\"xG,$*&%\"AG\"\"\"-%$expG6#,$*&%'lambdaGF(%\"tGF(!\"\"F(F0
" }{TEXT -1 53 ", and the gradient of both curves at these points is \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt = A*lambda*
exp(-lambda*t);" "6#/*&%#dxG\"\"\"%#dtG!\"\"*(%\"AGF&%'lambdaGF&-%$exp
G6#,$*&F+F&%\"tGF&F(F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
61 "These observations lend further support for the terminology. " }}
{PARA 0 "" 0 "" {TEXT -1 45 "In the following picture curves to repres
ent " }{XPPEDIT 18 0 "x=A*exp(-lambda*t)" "6#/%\"xG*&%\"AG\"\"\"-%$exp
G6#,$*&%'lambdaGF'%\"tGF'!\"\"F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "
x=-A*exp(-lambda*t)" "6#/%\"xG,$*&%\"AG\"\"\"-%$expG6#,$*&%'lambdaGF(%
\"tGF(!\"\"F(F0" }{TEXT -1 14 " are drawn in " }{TEXT 256 4 "blue" }
{TEXT -1 29 ", while a curve to represent " }{XPPEDIT 18 0 "x = A*exp(
-lambda*t)*sin(omega[0]*t+phi)" "6#/%\"xG*(%\"AG\"\"\"-%$expG6#,$*&%'l
ambdaGF'%\"tGF'!\"\"F'-%$sinG6#,&*&&%&omegaG6#\"\"!F'F.F'F'%$phiGF'F'
" }{TEXT -1 13 " is drawn in " }{TEXT 260 3 "red" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 1 " " }
{GLPLOT2D 612 331 331 {PLOTDATA 2 "6;-%'CURVESG6$7^r7$$\"\"!F)$\"3iua'
=\"y1rq!#=7$$\"3GLLL3x&)*3\"F,$\"3]TXr9\\/^#)F,7$$\"3emmm;arz@F,$\"3yW
Xbx%*z%**)F,7$$\"3;L$e*)4bQl#F,$\"3]&p*R%**3t<*F,7$$\"3v***\\7y%*z7$F,
$\"3)GVz'3p)QF*F,7$$\"3/LeRAY1lLF,$\"3=z)))4.b,H*F,7$$\"3Lm;ajW8-OF,$
\"3!\\#*yMQz_G*F,7$$\"3=+vo/V?RQF,$\"3q>5cO0Zf#*F,7$$\"3[LL$e9ui2%F,$
\"3'RC%))pV)H@*F,7$$\"3z***\\(oMrU^F,$\"3ml5T%f\">d()F,7$$\"3nmmm\"z_
\"4iF,$\"3L>.gll.GzF,7$$\"3Unmmm6m#G(F,$\"3!4\"HJkBrpnF,7$$\"39ommT&ph
N)F,$\"3'4h+hsdFN&F,7$$\"37M$3-js.*))F,$\"3!zo)4GeNuXF,7$$\"3A,+v=ddC%
*F,$\"3IX[F\"*>GgPF,7$$\"3?n;H2)y(e**F,$\"3;=Y^+uc?HF,7$$\"3KLLe*=)H\\
5!#<$\"3g]l2kLKl?F,7$$\"3wmm;ac#))4\"Fep$\"3gLV@6HJn7F,7$$\"3-++v=JN[6
Fep$\"3R<zXe/JDZ!#>7$$\"3CLLL$e!)y>\"Fep$!3G/pc<A$H6$Fbq7$$\"3smm\"z/3
uC\"Fep$!3/qdJE**ow5F,7$$\"3ULLe*ot*\\8Fep$!3Ev`Y+kxsDF,7$$\"3!****\\7
LRDX\"Fep$!3ZG_&yZv>!RF,7$$\"3[LLekGhe:Fep$!3!zY.P;Z'\\]F,7$$\"3%om;zR
'ok;Fep$!3nR'f/([vDfF,7$$\"3OLL3_(>/x\"Fep$!3nYU>g7Q,lF,7$$\"33++D1J:w
=Fep$!3=1$*=oitmnF,7$$\"3l;/,WIM.>Fep$!3%)y&z'\\vk%y'F,7$$\"3VL3x\")H`
I>Fep$!3st?3,f@#y'F,7$$\"3A]7`>Hsd>Fep$!3AGlN%fo'fnF,7$$\"3+n;HdG\"\\)
>Fep$!3\"f>QH=!H<nF,7$$\"37+D\"Gt#HR?Fep$!3Egk3HdXulF,7$$\"3oLLL3En$4#
Fep$!3Mt*y>CvqN'F,7$$\"3_++Dc#o%*=#Fep$!3?aZ/\"Gif!eF,7$$\"3#pmmT!RE&G
#Fep$!3QN=W_!R]1&F,7$$\"3eLLe9r5$R#Fep$!31p'zyi*QYSF,7$$\"3D+++D.&4]#F
ep$!3o9vZwWR&)GF,7$$\"3?+++]jB4EFep$!3K9)>-n[gj\"F,7$$\"3;+++vB_<FFep$
!3Gnqb(*47TOFbq7$$\"3!******\\-w=#GFep$\"3w9/l.D!=F)Fbq7$$\"33+++v'Hi#
HFep$\"3N_MbW\"eU$>F,7$$\"3pL$eky#*4-$Fep$\"32_2^;9ZHGF,7$$\"3&om;z*ev
:JFep$\"3)))*4,6.'4f$F,7$$\"3>+]7.%Q%GKFep$\"3-F$e_`(H#H%F,7$$\"3_LLL3
47TLFep$\"3yG:<'=\\$[ZF,7$$\"3WLLeR$\\))Q$Fep$\"35c6y#HyT'[F,7$$\"3#QL
L3xxlV$Fep$\"3g\"p;y'*4N$\\F,7$$\"3wLL3-iI%[$Fep$\"3G(z2#f>fc\\F,7$$\"
3nLLLLY.KNFep$\"3eTjK6')3M\\F,7$$\"3mm;HdO2VOFep$\"3t4)o+\\3?r%F,7$$\"
33++D\"o7Tv$Fep$\"3/7)HQ-]4F%F,7$$\"3km;HK5S_QFep$\"3o6`(zc\\Cs$F,7$$
\"3?LLL$Q*o]RFep$\"3_Nf?bR4`IF,7$$\"39n;H#GF&eSFep$\"31V:Uk-g:AF,7$$\"
3m++D\"=lj;%Fep$\"3h3s&zl$)>J\"F,7$$\"33++]iB0pUFep$\"3SL)=iO<[I%Fbq7$
$\"3S++vV&R<P%Fep$!3\"3K[5A'\\MVFbq7$$\"3Kn;zWG))yWFep$!3mWj.Lx3y7F,7$
$\"3CML$e9Ege%Fep$!3ro-*fq#\\H?F,7$$\"3'QL3F9<Wo%Fep$!3$G.f?V;:h#F,7$$
\"3]LLeR\"3Gy%Fep$!3)4Dwf/^72$F,7$$\"3/+](=7O*))[Fep$!3%R4(3`[Q:MF,7$$
\"3emm;/T1&*\\Fep$!3UNRe?Mj$f$F,7$$\"3(eT&)3RBE-&Fep$!3^_r,F1\\7OF,7$$
\"31mTgxE=]]Fep$!3,'*H0)3?,i$F,7$$\"3C;HKk>ux]Fep$!3r^>V>;i;OF,7$$\"3V
m;/^7I0^Fep$!3qK!f<*y7-OF,7$$\"3!o;zW#)>/;&Fep$!33z,\"o$\\%3a$F,7$$\"3
=nm\"zRQb@&Fep$!3yJ$o%3%ezV$F,7$$\"3nLLLe,]6`Fep$!3yxZ37Z@mJF,7$$\"3:+
+v=>Y2aFep$!3[pgz29]*y#F,7$$\"3Znm;zXu9cFep$!3qCN)[+8Xq\"F,7$$\"34+++]
y))GeFep$!3(HcPLH$4IQFbq7$$\"3H++]i_QQgFep$\"3<B4b#fmFr)Fbq7$$\"3b++D
\"y%3TiFep$\"3g-&)p*4N;&=F,7$$\"3G+]P4kh`jFep$\"3fpPJX\\GVAF,7$$\"3+++
]P![hY'Fep$\"3dEv!RE_f]#F,7$$\"3rK$eRP0n^'Fep$\"3#R(Ho_*Q(zDF,7$$\"3Km
mT5FEnlFep$\"3R7D_B#*pDEF,7$$\"3cLeky8a#f'Fep$\"3#zW`cug#QEF,7$$\"3#**
*\\(o/?yh'Fep$\"3?;SIA2#Rk#F,7$$\"3FmT5:()4VmFep$\"3=^(G:0MFk#F,7$$\"3
iKLL$Qx$omFep$\"3uJ@dhExMEF,7$$\"3:***\\78eBs'Fep$\"39p`M.>_&f#F,7$$\"
3cmm;z)Qjx'Fep$\"3o&f8Ci6p_#F,7$$\"3'RL$3F'>.$oFep$\"3?eO!*=sOICF,7$$
\"3Y+++v.I%)oFep$\"3EXY_['3wI#F,7$$\"3ML$ek`H@)pFep$\"38(3$Q[s#\\-#F,7
$$\"3?mm\"zpe*zqFep$\"3u(>B#e$fln\"F,7$$\"3;,++D\\'QH(Fep$\"3h3oya\"p4
k(Fbq7$$\"3%HL$e9S8&\\(Fep$!3I;f#=Ss4^\"Fbq7$$\"3s++D1#=bq(Fep$!3]hG)
\\[I\\+\"F,7$$\"3#om\"H2FO3yFep$!3W'evWay?M\"F,7$$\"3\"HLL$3s?6zFep$!3
Ad#osU`'4;F,7$$\"3yl;zpe()=!)Fep$!3aNxc(*[=1=F,7$$\"3a***\\7`Wl7)Fep$!
3sECh8'f>\">F,7$$\"3mlT5!R$Ry\")Fep$!3La\\\"fuf+$>F,7$$\"3cL$e*[ACI#)F
ep$!38B$\\h5hq#>F,7$$\"3Y,D\"y5\"4#G)Fep$!3:<)[(p*GM!>F,7$$\"3enmmm*RR
L)Fep$!3cY_l[1%)f=F,7$$\"3wnmTge)*R%)Fep$!3WWs5;AV7<F,7$$\"3%zmmTvJga)
Fep$!3qxEs#Gg_\\\"F,7$$\"3]MLe9tOc()Fep$!3/\"3qy!)HC4*Fbq7$$\"31,++]Qk
\\*)Fep$!3x)\\W(G[TSFFbq7$$\"3![LL3dg6<*Fep$\"3g6UJPa>'Q%Fbq7$$\"3%ymm
mw(Gp$*Fep$\"3!eSC2)[3(e*Fbq7$$\"3/M$eRA5\\Z*Fep$\"3Ojz(\\h()R;\"F,7$$
\"3C++D\"oK0e*Fep$\"3KXP;*H)f48F,7$$\"3d**\\il(z5j*Fep$\"3gK)y'o\"4pN
\"F,7$$\"3m+++]oi\"o*Fep$\"3+)e&37$>%*Q\"F,7$$\"3y,]PMR<K(*Fep$\"3cNe*
f0$429F,7$$\"35,+v=5s#y*Fep$\"3_hHh<p/59F,7$$\"3W,]P40O\"*)*Fep$\"3SEC
=jx+o8F,7$$\"#5F)$\"3y\"RZg&o<k7F,-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(
-F$6%7S7$F($\"\"\"F)7$F3$\"3S)4,[kAMd*F,7$FQ$\"3UvK:<A4<#*F,7$Fen$\"3#
41(\\y9=K))F,7$F_o$\"3#[MN]/_4Y)F,7$Fcp$\"3z3\"*[#H!)p5)F,7$Fiq$\"3!Qd
\\f5Y?z(F,7$Fcr$\"3cRE\\ik$)yuF,7$F]s$\"3%o$z!p^^\"orF,7$Fgs$\"3!**>Tb
$zIroF,7$Feu$\"3)[P0mC)yylF,7$F_v$\"3@vqe6#p9L'F,7$Fiv$\"3y7Z</R:kgF,7
$Fcw$\"3;7(=)**f42eF,7$F]x$\"31.nGo')opbF,7$Fgx$\"3;duQh%>DO&F,7$Fay$
\"3!4$[#e'3=E^F,7$Fez$\"3w8KF&))QT$\\F,7$F_[l$\"3n9%y(Q<y>ZF,7$Fi[l$\"
3g\"*Q))\\A#y`%F,7$Fc\\l$\"3#fJ:I9WiM%F,7$F]]l$\"33]#HSSR8<%F,7$Fg]l$
\"37*zIHoVj*RF,7$Fa^l$\"3;qRQ,o6UQF,7$F[_l$\"39e//MyU#o$F,7$Fi`l$\"3r;
7oI'zN_$F,7$Fcal$\"3md)zy'4*3R$F,7$Fhal$\"3QApe-/>`KF,7$F]bl$\"3t0Z+Y@
!o6$F,7$Fbbl$\"3O\"HpA\"y!*))HF,7$Fgbl$\"3gt&G:.h,(GF,7$Facl$\"3WoDZ0?
$Qu#F,7$F_el$\"3Bu3Rb'p]j#F,7$Fcfl$\"3OMr>BtpBDF,7$F]gl$\"3%)\\-FR\"[o
U#F,7$Fbgl$\"3o%>u,29_K#F,7$Fggl$\"3*yk%oaTZLAF,7$F\\hl$\"3l%pG6KY99#F
,7$Ffhl$\"3%z/o1MR^0#F,7$F`il$\"3]m,^t&3&o>F,7$Fdjl$\"3%=Q.e)o_))=F,7$
F^[m$\"3\"el&zNR45=F,7$Fc[m$\"3_U1B'*y_N<F,7$Fh[m$\"3Y+\\jE1sp;F,7$F]
\\m$\"3W\\#\\y8htf\"F,7$Fb\\m$\"3GN*>$oEIN:F,7$F\\]m$\"3arKyO))yr9F,7$
F`^m$\"3GTP'*4/Y89F,7$Fj^m$\"3-FhOKGN`8F,-F__m6&Fa_mF(F(Fb_m-%*LINESTY
LEG6#\"\"#-F$6%7S7$F($!\"\"F)7$F3$!3S)4,[kAMd*F,7$FQ$!3UvK:<A4<#*F,7$F
en$!3#41(\\y9=K))F,7$F_o$!3#[MN]/_4Y)F,7$Fcp$!3z3\"*[#H!)p5)F,7$Fiq$!3
!Qd\\f5Y?z(F,7$Fcr$!3cRE\\ik$)yuF,7$F]s$!3%o$z!p^^\"orF,7$Fgs$!3!**>Tb
$zIroF,7$Feu$!3)[P0mC)yylF,7$F_v$!3@vqe6#p9L'F,7$Fiv$!3y7Z</R:kgF,7$Fc
w$!3;7(=)**f42eF,7$F]x$!31.nGo')opbF,7$Fgx$!3;duQh%>DO&F,7$Fay$!3!4$[#
e'3=E^F,7$Fez$!3w8KF&))QT$\\F,7$F_[l$!3n9%y(Q<y>ZF,7$Fi[l$!3g\"*Q))\\A
#y`%F,7$Fc\\l$!3#fJ:I9WiM%F,7$F]]l$!33]#HSSR8<%F,7$Fg]l$!37*zIHoVj*RF,
7$Fa^l$!3;qRQ,o6UQF,7$F[_l$!39e//MyU#o$F,7$Fi`l$!3r;7oI'zN_$F,7$Fcal$!
3md)zy'4*3R$F,7$Fhal$!3QApe-/>`KF,7$F]bl$!3t0Z+Y@!o6$F,7$Fbbl$!3O\"HpA
\"y!*))HF,7$Fgbl$!3gt&G:.h,(GF,7$Facl$!3WoDZ0?$Qu#F,7$F_el$!3Bu3Rb'p]j
#F,7$Fcfl$!3OMr>BtpBDF,7$F]gl$!3%)\\-FR\"[oU#F,7$Fbgl$!3o%>u,29_K#F,7$
Fggl$!3*yk%oaTZLAF,7$F\\hl$!3l%pG6KY99#F,7$Ffhl$!3%z/o1MR^0#F,7$F`il$!
3]m,^t&3&o>F,7$Fdjl$!3%=Q.e)o_))=F,7$F^[m$!3\"el&zNR45=F,7$Fc[m$!3_U1B
'*y_N<F,7$Fh[m$!3Y+\\jE1sp;F,7$F]\\m$!3W\\#\\y8htf\"F,7$Fb\\m$!3GN*>$o
EIN:F,7$F\\]m$!3arKyO))yr9F,7$F`^m$!3GTP'*4/Y89F,7$Fj^m$!3-FhOKGN`8F,F
[imF]im-F$6%7$7$$\"31+++sN*)[FFepF(7$F[cn$!3A+++++++!*F,-F__m6&Fa_mF)F
)F)F]im-F$6%7$7$$\"38+++O<HMNFepF(7$Ffcn$\"3a**************pF,F`cnF]im
-F$6%7$7$$\"36+++**)*o>VFepF(7$F_dnF^cnF`cnF]im-F$6%7$7$$\"3A+++!R%)en
'FepF(7$FfdnFicnF`cnF]im-F$6'7$7$$\"#bFfim$\"\"'Ffim7$$\"+!R%)en'!\"*F
_en7%7$$\"+^fHelFden$\"+++++j!#5Faen7$Fgen$\"+++++dF[fn-%&STYLEG6#%,PA
TCHNOGRIDGF`cn-F^im6#\"\"$-F$6'7$7$$\"#ZFfimF_en7$$\"+O<HMNFdenF_en7%7
$$\"+iD'3l$FdenF]fnF\\gn7$FagnFienF_fnF`cnFcfn-F$6'7$7$$\"$D$!\"#$Fd_m
Ffim7$$\"+sN*)[FFdenF[hn7%7$$\"+e[6\\GFden$!+++++$)F[fnF\\hn7$Fahn$!++
+++xF[fnF_fnF`cnFcfn-F$6'7$7$$\"$v$FjgnF[hn7$$\"+**)*o>VFdenF[hn7%7$$
\"+>>v0UFdenFfhnF^in7$FcinFchnF_fnF`cnFcfn-%%TEXTG6$7$Fj^m$!\"&FjgnQ\"
t6\"-Fgin6$7$$!#:Fjgn$\"#6FfimQ\"xF]jn-Fgin6$7$$\"#^Ffim$\"#mFjgnQ$___
F]jn-Fgin6$7$$\"$`$Fjgn$!#uFjgnQ#__F]jn-Fgin6$7$$\"$-&Fjgn$\"#pFjgnQ\"
2F]jn-Fgin6$7$$\"$E&Fjgn$\"#_FjgnQ\"oF]jn-Fgin6$7$$\"#PFfim$!#))FjgnFe
\\o-Fgin6%7$$\"$>&Fjgn$\"\"(FfimQ\"pF]jn-%%FONTG6$%'SYMBOLG\"#7-Fgin6%
7$Fijn$F^enFjgnQ\"wF]jnFe]o-Fgin6%7$Fa[o$!\"(FfimFd]oFe]o-Fgin6%7$Fa[o
$!#&)FjgnF^^oFe]o-%*AXESTICKSG6$F)F)-%+AXESLABELSG6%F\\jnQ!F]jn-Ff]o6#
%(DEFAULTG-%%VIEWG6$;FajnFj^m;$!#6FfimFcjn" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve
 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17
" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" }}{TEXT -1 1 
" " }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Summary " }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 83 "Given an oscillatory system whose motion is governed by the dif
ferential equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "d^2*x/(d*t^2)+2*lambda;" "6#,&*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF
&F(!\"\"F(*&F&F(%'lambdaGF(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+o
mega^2*x = 0;" "6#/,&*&%#dxG\"\"\"%#dtG!\"\"F'*&%&omegaG\"\"#%\"xGF'F'
\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "
" 0 "" {TEXT -1 14 "the motion is " }{TEXT 259 12 "under-damped" }
{TEXT -1 6 " when " }{XPPEDIT 18 0 "0<lambda" "6#2\"\"!%'lambdaG" }
{XPPEDIT 18 0 "``<omega" "6#2%!G%&omegaG" }{TEXT -1 2 ", " }{TEXT 259 
11 "over-damped" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "omega<lambda" "6
#2%&omegaG%'lambdaG" }{TEXT -1 6 ", and " }{TEXT 259 17 "critically-da
mped" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "lambda=omega" "6#/%'lambdaG
%&omegaG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 58 "The situatio
n is represented pictorially by the following " }{XPPEDIT 18 0 "lambda
" "6#%'lambdaG" }{TEXT -1 14 " number line. " }}{PARA 0 "" 0 "" {TEXT 
-1 1 " " }{GLPLOT2D 743 141 141 {PLOTDATA 2 "65-%'CURVESG6%7$7$$\"\"!F
)F(7$$\"\"(F)F(-%'COLOURG6&%$RGBG$\")!\\DP\"!\")F1$\")viobF3-%*THICKNE
SSG6#\"\"#-F$6&7$F'7$F($\"3++++++++]!#=-F.6&F0F(F($\"*++++\"F3-%&STYLE
G6#%%LINEGF6-F$6&7$7$$\"\"%F)F(7$FMF>FAFEF6-F$6%7$FL7$FM$!3++++++++]F@
-F.6&F0F)F)F)-%*LINESTYLEGF8-F$6%7$F'7$F(FTFVFX-F$6'7$7$$\"$X\"!\"#$!
\"$!\"\"7$F(F_o7%7$$\")+++()!\"*$!+++++N!#5Fbo7$Feo$!+++++DFjo-FF6#%,P
ATCHNOGRIDG-F.6&F0$\")#)eqkF3$\"))eqk\"F3Fep-FY6#\"\"&-F$6'7$7$$\"$b#F
^oF_o7$$\"$+%F^oF_o7%7$$\"++++8RFgoF\\pF`q7$FeqFhoF^pFapFgp-F$6'7$7$$F
ipF)F_o7$FMF_o7%7$$\"++++!4%FgoFhoF]r7$F`rF\\pF^pFapFgp-F$6'7$7$$\"\"'
F)F_o7$F+F_o7%7$$\"++++5pFgoF\\pFir7$F\\sFhoF^pFapFgp-%%TEXTG6%7$F($F,
FaoQ\"06\"-%%FONTG6$%'SYMBOLG\"#7-F`s6%7$FMFcsQ\"wFesFfs-F`s6%7$$\"#sF
aoF(Q\"lFesFfs-F`s6%7$$F9F)F_oQ-under-dampedFes-Fgs6$%*HELVETICAG\"#5-
F`s6%7$$\"#bFaoF_oQ,over-dampedFesFjt-F`s6%7$FM$!#jF^oQ+criticallyFesF
jt-F`s6%7$FM$FgoFaoQ'dampedFesFjt-%+AXESLABELSG6%Q!FesFbv-Fgs6#%(DEFAU
LTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$FevFev" 1 2 0 1 10 0 2 9 1 1 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve
 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" }}{TEXT 
-1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 122 "A weight attached \+
to a spring experiences a resistance to motion so that the differentia
l equation governing its motion is" }}{PARA 256 "" 0 "" {TEXT -1 2 "  \+
" }{XPPEDIT 18 0 "d^2*x/(d*t^2)+8;" "6#,&*(%\"dG\"\"#%\"xG\"\"\"*&F%F(
*$%\"tGF&F(!\"\"F(\"\")F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+16*x \+
= 0;" "6#/,&*&%#dxG\"\"\"%#dtG!\"\"F'*&\"#;F'%\"xGF'F'\"\"!" }{TEXT 
-1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 274 1 "x" }
{TEXT -1 71 " is the displacement downward from the equilibrium positi
on in cm. and " }{TEXT 273 1 "t" }{TEXT -1 21 " is the time in secs." 
}}{PARA 0 "" 0 "" {TEXT -1 112 "When first observed the weight is 1 cm
. above the equilibrium position and has a velocity of 8 cm/sec downwa
rds." }}{PARA 0 "" 0 "" {TEXT -1 34 "This gives the initial conditions
 " }{XPPEDIT 18 0 "x(0) = -1" "6#/-%\"xG6#\"\"!,$\"\"\"!\"\"" }{TEXT 
-1 5 " and " }{XPPEDIT 18 0 "eval(dx/dt,t = 0) = 8;" "6#/-%%evalG6$*&%
#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!\"\")" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "(a) Find an expression \+
for the displacement " }{TEXT 275 1 "t" }{TEXT -1 42 " secs. after the
 weight is first observed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 54 "(b) Plot a graph of the displacement against time
 for " }{TEXT 276 1 "t" }{TEXT -1 21 " between 0 to 2 secs." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "(c) Find the ti
me when the weight first passes through the equilibrium position." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "(d) Find \+
the maximum displacement of the weight downwards from the equilibrium \+
position." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
74 "(e) Classify the motion as under-damped, over-damped or critically
 damped." }}{PARA 0 "" 0 "" {TEXT -1 48 " ____________________________
___________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 47 "_______________________________________
________" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 122 "A weight
 attached to a spring experiences a resistance to motion so that the d
ifferential equation governing its motion is" }}{PARA 256 "" 0 "" 
{TEXT -1 4 "  4 " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+dx/dt+2*x = 0;" "6#/,(
*(%\"dG\"\"#%\"xG\"\"\"*&F&F)*$%\"tGF'F)!\"\"F)*&%#dxGF)%#dtGF-F)*&F'F
)F(F)F)\"\"!" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }
{TEXT 280 1 "x" }{TEXT -1 71 " is the displacement downward from the e
quilibrium position in cm. and " }{TEXT 281 1 "t" }{TEXT -1 21 " is th
e time in secs." }}{PARA 0 "" 0 "" {TEXT -1 130 "The weight is release
d from a point 5 cm. below the equilibrium position (with zero velocit
y), which gives the initial conditions " }{XPPEDIT 18 0 "x(0) = 5" "6#
/-%\"xG6#\"\"!\"\"&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "eval(dx/dt,t \+
= 0) = 0;" "6#/-%%evalG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!F." }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
44 "(a) Find an expression for the displacement " }{TEXT 277 1 "t" }
{TEXT -1 36 " secs. after the weight is released." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "(b) Plot a graph of the d
isplacement against time for " }{TEXT 278 1 "t" }{TEXT -1 22 " between
 0 to 12 secs." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 80 "(c) Find the time when the weight first passes through th
e equilibrium position." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 86 "(d) Find the maximum displacement of the weight upwa
rds from the equilibrium position." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 74 "(e) Classify the motion as under-damped, \+
over-damped or critically damped." }}{PARA 0 "" 0 "" {TEXT -1 48 " ___
____________________________________________" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 47 "____________
___________________________________" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{SECT 1 {PARA 0 "" 0 "" {TEXT -1 25 "Code for drawing pictures" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "
" {TEXT -1 27 "Code for 1st spring picture" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 527 "PLOT(CURVES([[0,.
5],[.05,.5],[.1,.8],[.2,.2],[.3,.8],[.4,.2],\n[.5,.8],[.6,.2],[.7,.8],
[.8,.2],[.9,.8],[1,.2],\n[1.1,.8],[1.2,.2],[1.3,.8],[1.4,.2],[1.5,.8],
[1.6,.2],\n[1.7,.8],[1.8,.2],[1.9,.8],[2,.2],[2.1,.8],[2.2,.2],\n[2.3,
.8],[2.4,.2],[2.5,.8],[2.6,.2],[2.7,.8],[2.8,.2],\n[2.9,.8],[3,.2],[3.
05,.5],[3.1,.5]],\nTHICKNESS(2),COLOR(RGB,0,.6,0)),\nCURVES([[0,1],[0,
0],[4,0]],THICKNESS(2),COLOR(RGB,.2,.2,.2)),\nPOLYGONS([[3.1,.02],[3.1
,.98],[3.6,.98],[3.6,.02],[3.1,.02]],\nCOLOR(RGB,.8,.8,.8)),\nTEXT([3.
35,.5],`M`),\nAXESSTYLE(NONE));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "
" {TEXT -1 27 "Code for 2nd spring picture" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 677 "PLOT(CURVES([[0,.
5],[.05,.5],[.1,.8],[.2,.2],[.3,.8],[.4,.2],\n[.5,.8],[.6,.2],[.7,.8],
[.8,.2],[.9,.8],[1,.2],\n[1.1,.8],[1.2,.2],[1.3,.8],[1.4,.2],[1.5,.8],
[1.6,.2],\n[1.7,.8],[1.8,.2],[1.9,.8],[2,.2],[2.1,.8],[2.2,.2],\n[2.3,
.8],[2.4,.2],[2.5,.8],[2.6,.2],[2.7,.8],[2.8,.2],\n[2.9,.8],[3,.2],[3.
05,.5],[3.1,.5]],\nTHICKNESS(2),COLOR(RGB,0,.6,0)),\nCURVES([[0,1],[0,
0],[5,0]],THICKNESS(2),COLOR(RGB,.2,.2,.2)),\nPOLYGONS([[3.1,.02],[3.1
,.98],[3.6,.98],[3.6,.02],[3.1,.02]],\nCOLOR(RGB,.8,.8,.8)),\nPOLYGONS
([[4.5,0],[4.5,1],[5,1],[5,0],[4.5,0]],\nCOLOR(RGB,.8,.5,0)),\nCURVES(
[[4,1],[5,1],[5,0]],[[3.6,.5],[4.5,.5]],[[4.5,0],[4.5,1]],\nTHICKNESS(
2)),\nTEXT([3.35,.5],`M`),\nAXESSTYLE(NONE));" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 
1 {PARA 0 "" 0 "" {TEXT -1 40 "Code for under-damped oscillations grap
h" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1024 "g := t -> exp(-t/5): \np1 := plot(exp(-t/5)*sin(2*t
+Pi/4),t=0..10,color=red):\np2 := plot([exp(-t/5),-exp(-t/5)],t=0..10,
color=blue,linestyle=2):\na := evalf(7*Pi/8): b := evalf(9*Pi/8): c :=
 evalf(11*Pi/8): d := evalf(17*Pi/8):  \np3 := plot([[[a,0],[a,-.9]],[
[b,0],[b,.7]],[[c,0],[c,-.9]],\n    [[d,0],[d,.7]]],color=black,linest
yle=2):\np4 := plottools[arrow]([5.5,.6],[d,.6],0,.06,.1,arrow,color=b
lack,linestyle=3):\np5 := plottools[arrow]([4.7,.6],[b,.6],0,.06,.1,ar
row,color=black,linestyle=3):\np6 := plottools[arrow]([3.25,-.8],[a,-.
8],0,.06,.2,arrow,color=black,linestyle=3):\np7 := plottools[arrow]([3
.75,-.8],[c,-.8],0,.06,.2,arrow,color=black,linestyle=3):\nt1 := plots
[textplot]([[10,-.05,`t`],[-.15,1.1,`x`],\n         [5.1,.66,`___`],[3
.53,-.74,`__`],[5.02,.69,`2`]]):\nt2 := plots[textplot]([[5.26,.52,`o`
],[3.7,-.88,`o`]]):\nt3 := plots[textplot]([[5.19,.7,`p`],[5.1,.55,`w`
],[3.53,-.7,`p`],[3.53,-.85,`w`]\n             ],font=[SYMBOL,12]):\np
lots[display]([p||(1..7),t1,t2,t3],tickmarks=[0,0],view=[-.15..10,-1.1
..1.1]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 24 "Code for s
ummary picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 831 "p1 := plot([[0,0],[7,0]],thickness=2,color=navy
):\np2 := plot([[[0,0],[0,1/2]],[[4,0],[4,1/2]]],style=line,color=blue
,thickness=2):\np3 := plot(\{[[0,0],[0,-1/2]],[[4,0],[4,-1/2]]\},color
=black,linestyle=2):\np4 := plottools[arrow]([1.45,-.3],[0,-.3],0,.1,.
06,arrow,color=brown,linestyle=5):\np5 := plottools[arrow]([2.55,-.3],
[4,-.3],0,.1,.06,arrow,color=brown,linestyle=5):\np6 := plottools[arro
w]([5,-.3],[4,-.3],0,.1,.09,arrow,color=brown,linestyle=5):\np7 := plo
ttools[arrow]([6,-.3],[7,-.3],0,.1,.09,arrow,color=brown,linestyle=5):
\nt1 := plots[textplot](\{[0,.7,`0`],[4,.7,`w`]\},font=[SYMBOL,12]):\n
t2 := plots[textplot]([7.2,0,`l`],font=[SYMBOL,12]):\nt3 := plots[text
plot]([[2,-.3,`under-damped`],[5.5,-.3,`over-damped`],\n  [4,-.63,`cri
tically`],[4,-.9,`damped`]],font=[HELVETICA,10]):\nplots[display]([p||
(1..7),t||(1..3)],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 
1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }