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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 69 "Second order linear differential \+
equations with constant coefficients" }}{PARA 0 "" 0 "" {TEXT -1 37 "b
y Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "V
ersion:  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 42 "Second orde
r linear differential equations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 47 "A second order differential \+
equation is called " }{TEXT 259 6 "linear" }{TEXT -1 33 " if it can be
 written in the form" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "f(x);" "6#-%
\"fG6#%\"xG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+g(x);" "6#,
&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(-%\"gG6#F+F(" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "dy/dx+h(x)*y = k(x);" "6#/,&*&%#dyG\"\"\"%#dx
G!\"\"F'*&-%\"hG6#%\"xGF'%\"yGF'F'-%\"kG6#F." }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 17 "If the functions " }{XPPEDIT 18 0 "f(x), \+
g(x)" "6$-%\"fG6#%\"xG-%\"gG6#F&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "
h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 5 " are " }{TEXT 259 8 "constant" }
{TEXT -1 52 ", then such a differential equation is said to have " }
{TEXT 259 21 "constant coefficients" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 17 "If, in addition, " }{XPPEDIT 18 0 "k(x)" "6#-%\"kG6#%\"
xG" }{TEXT -1 32 " is zero, then it is said to be " }{TEXT 259 11 "hom
ogeneous" }{TEXT -1 2 ".\n" }}{PARA 0 "" 0 "" {TEXT -1 25 "Examples of
 second order " }{TEXT 259 6 "linear" }{TEXT -1 24 " differential equa
tions." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 
2 "  " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+sin*x;" "6#,&*(%\"dG\"\"#%\"yG\"
\"\"*&F%F(*$%\"xGF&F(!\"\"F(*&%$sinGF(F+F(F(" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "dy/dx+x^3*y = exp(-x)+2*x;" "6#/,&*&%#dyG\"\"\"%#dxG!\"
\"F'*&%\"xG\"\"$%\"yGF'F',&-%$expG6#,$F+F)F'*&\"\"#F'F+F'F'" }{TEXT 
-1 13 "  ------- (i)" }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "x^2;" "6#*$%\"xG
\"\"#" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+x;" "6#,&*(%\"dG
\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(F+F(" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "dy/dx+(x^2-4)*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&
,&*$%\"xG\"\"#F'\"\"%F)F'%\"yGF'F'\"\"!" }{TEXT -1 14 "  ------- (ii)
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {XPPEDIT 18 0 "2;" "6#\"\"#" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "d^2*y/(d*x^2)+5;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%
\"xGF&F(!\"\"F(\"\"&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+3*y = x^
2+cos*x;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"$F'%\"yGF'F',&*$%\"xG\"
\"#F'*&%$cosGF'F/F'F'" }{TEXT -1 15 "  ------- (iii)" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{XPPEDIT 18 0 "2;" "6#\"\"#" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/(d*
x^2)+5;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(\"\"&F(" 
}{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+3*y = 0;" "6#/,&*&%#dyG\"\"\"%#d
xG!\"\"F'*&\"\"$F'%\"yGF'F'\"\"!" }{TEXT -1 14 "  ------- (iv)" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "(iii) and
 (iv) have constant coefficients, and (iv) is homogeneous." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 64 "Non-homo
geneous 2nd order linear DE's with constant coefficients" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Le
t " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "d^2*y/(d*x^2)+b;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%
\"xGF&F(!\"\"F(%\"bGF(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+c*y = f(
x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&%\"cGF'%\"yGF'F'-%\"fG6#%\"xG" }
{TEXT -1 14 "   ------- (N)" }}{PARA 0 "" 0 "" {TEXT -1 73 "be a a 2nd
 order linear differential equation with constant coefficients." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Suppose t
hat we can obtain a general solution " }{XPPEDIT 18 0 "y(x) = c(x);" "
6#/-%\"yG6#%\"xG-%\"cG6#F'" }{TEXT -1 8 " of the " }{TEXT 259 34 "corr
esponding homogeneous equation" }{TEXT -1 28 " with the same coefficie
nts." }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "d^2*y/(d*x^2)+b;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(
*$%\"xGF&F(!\"\"F(%\"bGF(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+c*y =
 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&%\"cGF'%\"yGF'F'\"\"!" }{TEXT 
-1 14 "   ------- (H)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 42 "In relation to the original equation (N), " }{XPPEDIT 
18 0 "c(x)" "6#-%\"cG6#%\"xG" }{TEXT -1 15 " is called the " }{TEXT 
259 22 "complementary solution" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Suppose also that, by som
e means, we can find a " }{TEXT 259 19 "particular solution" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "y(x) = p(x);" "6#/-%\"yG6#%\"xG-%\"pG6#F'" }
{TEXT -1 8 " of the " }{TEXT 259 33 "original non-homogeneous equation
" }{TEXT -1 5 " (N)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 9 "Then the " }{TEXT 259 16 "general solution" }{TEXT -1 
11 " of (N) is:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "y(x) = c(x)+p(x);
" "6#/-%\"yG6#%\"xG,&-%\"cG6#F'\"\"\"-%\"pG6#F'F," }{TEXT -1 1 "." }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 262 10 "__________" }{TEXT -1 
2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "
To see why, consider the differential operator:  " }{XPPEDIT 18 0 "a*D
^2+b*D+c;" "6#,(*&%\"aG\"\"\"*$%\"DG\"\"#F&F&*&%\"bGF&F(F&F&%\"cGF&" }
{TEXT -1 8 ", where " }{XPPEDIT 18 0 "D = d/dx;" "6#/%\"DG*&%\"dG\"\"
\"%#dxG!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "D^2 = d^2/(d*x^2);" 
"6#/*$%\"DG\"\"#*&%\"dGF&*&F(\"\"\"*$%\"xGF&F*!\"\"" }{TEXT -1 1 "." }
}{PARA 0 "" 0 "" {TEXT -1 26 "Applying this operator to " }{XPPEDIT 
18 0 "c(x)" "6#-%\"cG6#%\"xG" }{TEXT -1 32 " gives zero, and applying \+
it to " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 7 " gives " 
}{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Thus, by linearity, \+
applying the differential operator  " }{XPPEDIT 18 0 "a*D^2+b*D+c;" "6
#,(*&%\"aG\"\"\"*$%\"DG\"\"#F&F&*&%\"bGF&F(F&F&%\"cGF&" }{TEXT -1 5 " \+
 to " }{XPPEDIT 18 0 "y(x) = c(x)+p(x);" "6#/-%\"yG6#%\"xG,&-%\"cG6#F'
\"\"\"-%\"pG6#F'F," }{TEXT -1 7 " gives " }{XPPEDIT 18 0 "0+f(x) = f(x
)" "6#/,&\"\"!\"\"\"-%\"fG6#%\"xGF&-F(6#F*" }{TEXT -1 11 ", that is, \+
" }{XPPEDIT 18 0 "y(x) = c(x)+p(x);" "6#/-%\"yG6#%\"xG,&-%\"cG6#F'\"\"
\"-%\"pG6#F'F," }{TEXT -1 22 " is a solution of (N)." }}{PARA 0 "" 0 "
" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "c(x)" "6#-%\"cG6#%\"xG" }{TEXT 
-1 43 " contains two arbitrary constants, so does " }{XPPEDIT 18 0 "y(
x) = c(x)+p(x);" "6#/-%\"yG6#%\"xG,&-%\"cG6#F'\"\"\"-%\"pG6#F'F," }
{TEXT -1 55 ", and therefore it must be the general solution of (N)." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 
50 "The method of undetermined coefficients: Example 1" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 272 8 "Q
uestion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 58 "(a) Find the \+
general solution of the differential equation" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y
/(d*x^2)+dy/dx-2*y = 2*x^2-10*x+2;" "6#/,(*(%\"dG\"\"#%\"yG\"\"\"*&F&F
)*$%\"xGF'F)!\"\"F)*&%#dyGF)%#dxGF-F)*&F'F)F(F)F-,(*&F'F)*$F,F'F)F)*&
\"#5F)F,F)F-F'F)" }{TEXT -1 15 "  ------- (i). " }}{PARA 0 "" 0 "" 
{TEXT -1 74 "(b) Find the particular solution of (i) subject to the in
itial conditions " }{XPPEDIT 18 0 "y(0) = 3;" "6#/-%\"yG6#\"\"!\"\"$" 
}{TEXT -1 5 " and " }{XPPEDIT 18 0 "y*`'`(0) = -2;" "6#/*&%\"yG\"\"\"-
%\"'G6#\"\"!F&,$\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 257 "" 0 "" {TEXT 261 8 "Solution" }{TEXT 273 2 ": " }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "(a) The solution of the c
orresponding homogeneous equation:" }}{PARA 256 "" 0 "" {TEXT -1 3 "  \+
 " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+dy/dx-2*y = 0;" "6#/,(*(%\"dG\"\"#%\"
yG\"\"\"*&F&F)*$%\"xGF'F)!\"\"F)*&%#dyGF)%#dxGF-F)*&F'F)F(F)F-\"\"!" }
{TEXT -1 15 " ------- (ii), " }}{PARA 0 "" 0 "" {TEXT -1 51 "is obtain
ed by considering the auxiliary equation: " }}{PARA 256 "" 0 "" {TEXT 
-1 2 "  " }{XPPEDIT 18 0 "m^2+m-2 = 0;" "6#/,(*$%\"mG\"\"#\"\"\"F&F(F'
!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "By factori
ng the left side we obtain:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "(m+2)*(m-1) = 0;" "6#/*&,&%\"mG\"\"\"\"\"#F'F',&F&F'F'!
\"\"F'\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 "Hence the
 roots of the auxiliary equation are " }{XPPEDIT 18 0 "m = -2;" "6#/%
\"mG,$\"\"#!\"\"" }{TEXT -1 6 "  and " }{XPPEDIT 18 0 "m = 1;" "6#/%\"
mG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 43 "The general \+
solution of (ii), which is the " }{TEXT 259 22 "complementary solution
" }{TEXT -1 12 " of (i) is: " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "c(x) = C[1]*exp(-2*x)+C[2]*exp(x);" "6#/-%\"cG6#%\"xG,&
*&&%\"CG6#\"\"\"F--%$expG6#,$*&\"\"#F-F'F-!\"\"F-F-*&&F+6#F3F--F/6#F'F
-F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 83 "deHG := diff(y(x),x$2)+diff(y(x),x)-2*y(x)=
0;\ndsolve(deHG,y(x)):\nsubs(y(x)=c(x),%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%deHGG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"
-F(6$F*F-F2*&F1F2F*F2!\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%
\"cG6#%\"xG,&*&%$_C1G\"\"\"-%$expGF&F+F+*&%$_C2GF+-F-6#,$*&\"\"#F+F'F+
!\"\"F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "It is likely that \+
we could find a particular solution of (i) having the form of a " }
{TEXT 259 20 "quadratic polynomial" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "p
(x) = A*x^2+B*x+C;" "6#/-%\"pG6#%\"xG,(*&%\"AG\"\"\"*$F'\"\"#F+F+*&%\"
BGF+F'F+F+%\"CGF+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 37 "In f
act we can find the coefficients " }{XPPEDIT 18 0 "A, B" "6$%\"AG%\"BG
" }{TEXT -1 5 " and " }{TEXT 275 1 "C" }{TEXT -1 33 " in a fairly stra
ightforward way." }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{TEXT 276 1 "
y" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "p(x) = A*x^2+B*x+C" "6#/-%\"pG6#%
\"xG,(*&%\"AG\"\"\"*$F'\"\"#F+F+*&%\"BGF+F'F+F+%\"CGF+" }{TEXT -1 11 "
, we have: " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "dy/dx
 = p*`'`(x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%\"pGF&-%\"'G6#%\"xGF&" }
{XPPEDIT 18 0 "`` = 2*A*x+B;" "6#/%!G,&*(\"\"#\"\"\"%\"AGF(%\"xGF(F(%
\"BGF(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 
256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = p*`''`(x);
" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"*&%\"pGF(-%#''G6#F+
F(" }{XPPEDIT 18 0 "`` = 2*A;" "6#/%!G*&\"\"#\"\"\"%\"AGF'" }{TEXT -1 
2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "Substituting the last three expr
essions for " }{XPPEDIT 18 0 "d^2*y/(d*x^2)" "6#*(%\"dG\"\"#%\"yG\"\"
\"*&F$F'*$%\"xGF%F'!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "dy/dx" "6#*
&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 5 " and " }{TEXT 280 1 "y" }{TEXT 
-1 28 " in (i) gives the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "2*A+``(2*A*x+B)-2*``(A*x^2+B*x+C) = 2*x^2-10*x+2;" "
6#/,(*&\"\"#\"\"\"%\"AGF'F'-%!G6#,&*(F&F'F(F'%\"xGF'F'%\"BGF'F'*&F&F'-
F*6#,(*&F(F'*$F.F&F'F'*&F/F'F.F'F'%\"CGF'F'!\"\",(*&F&F'*$F.F&F'F'*&\"
#5F'F.F'F8F&F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}
{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "2*A+2*A*x+B-2*A*x^2-
2*B*x-2*C = 2*x^2-10*x+2;" "6#/,.*&\"\"#\"\"\"%\"AGF'F'*(F&F'F(F'%\"xG
F'F'%\"BGF'*(F&F'F(F'F*F&!\"\"*(F&F'F+F'F*F'F-*&F&F'%\"CGF'F-,(*&F&F'*
$F*F&F'F'*&\"#5F'F*F'F-F&F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT 
-1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 18 0 "-
2*A*x^2+``(2*A-2*B)*x+``(2*A+B-2*C) = 2*x^2-10*x+2;" "6#/,(*(\"\"#\"\"
\"%\"AGF'%\"xGF&!\"\"*&-%!G6#,&*&F&F'F(F'F'*&F&F'%\"BGF'F*F'F)F'F'-F-6
#,(*&F&F'F(F'F'F2F'*&F&F'%\"CGF'F*F',(*&F&F'*$F)F&F'F'*&\"#5F'F)F'F*F&
F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 101 "We can compare coefficients on the left and right side
s of the last equation to obtain equations for " }{TEXT 277 1 "A" }
{TEXT -1 2 ", " }{TEXT 278 1 "B" }{TEXT -1 5 " and " }{TEXT 279 1 "C" 
}{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PI
ECEWISE([`coefficients of `*x^2*`  ..`, -2*A = 2],[`coefficients of `*
x*`  ..`, 2*A-2*B = -10],[`coefficients of `*x^0*`  ..`, 2*A+B-2*C = 2
]);" "6#-%*PIECEWISEG6%7$*(%1coefficients~of~G\"\"\"*$%\"xG\"\"#F)%%~~
..GF)/,$*&F,F)%\"AGF)!\"\"F,7$*(F(F)F+F)F-F)/,&*&F,F)F1F)F)*&F,F)%\"BG
F)F2,$\"#5F27$*(F(F)*$F+\"\"!F)F-F)/,(*&F,F)F1F)F)F9F)*&F,F)%\"CGF)F2F
," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 25 "From the first equation, " }{XPPEDIT 18 0 "A=-1" "6#/%\"A
G,$\"\"\"!\"\"" }{TEXT -1 20 ". Then substituting " }{XPPEDIT 18 0 "A=
-1" "6#/%\"AG,$\"\"\"!\"\"" }{TEXT -1 30 " in the second equation give
s " }{XPPEDIT 18 0 "-2-2*B=-10" "6#/,&\"\"#!\"\"*&F%\"\"\"%\"BGF(F&,$
\"#5F&" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "B=4" "6#/%\"BG\"\"%" }
{TEXT -1 32 ". The last equation now becomes " }{XPPEDIT 18 0 "-2+4-2*
C=2" "6#/,(\"\"#!\"\"\"\"%\"\"\"*&F%F(%\"CGF(F&F%" }{TEXT -1 9 " so th
at " }{XPPEDIT 18 0 "C=0" "6#/%\"CG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Hence (i) has the \+
" }{TEXT 259 19 "particular solution" }{TEXT -1 2 "  " }{XPPEDIT 18 0 
"p(x) = 4*x-x^2;" "6#/-%\"pG6#%\"xG,&*&\"\"%\"\"\"F'F+F+*$F'\"\"#!\"\"
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 107 "This method of finding a particular solution, once we ha
ve identified a general form to use, is called the " }{TEXT 259 35 "me
thod of undetermined coefficients" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 46 "The general solution of (i) is the sum of the " }{TEXT 
259 22 "complementary solution" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "c(x) \+
= C[1]*exp(-2*x)+C[2]*exp(x);" "6#/-%\"cG6#%\"xG,&*&&%\"CG6#\"\"\"F--%
$expG6#,$*&\"\"#F-F'F-!\"\"F-F-*&&F+6#F3F--F/6#F'F-F-" }{TEXT -1 9 " a
nd the " }{TEXT 259 19 "particular solution" }{TEXT -1 2 "  " }
{XPPEDIT 18 0 "p(x) = 4*x-x^2;" "6#/-%\"pG6#%\"xG,&*&\"\"%\"\"\"F'F+F+
*$F'\"\"#!\"\"" }{TEXT -1 9 ", namely:" }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "y = C[1]*exp(-2*x)+C[2]*exp(x)+4*x-x^2;" "6#/%\"
yG,**&&%\"CG6#\"\"\"F*-%$expG6#,$*&\"\"#F*%\"xGF*!\"\"F*F**&&F(6#F0F*-
F,6#F1F*F**&\"\"%F*F1F*F**$F1F0F2" }{TEXT -1 16 " ------- (iii). " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
133 "de := diff(y(x),x$2)+diff(y(x),x)-2*y(x)=2*x^2-10*x+2;\nunassign(
'A','B','C'):\npx := A*x^2+B*x+C;\nsubs(y(x)=px,de);\neq := simplify(%
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG
-%\"$G6$F-\"\"#\"\"\"-F(6$F*F-F2*&F1F2F*F2!\"\",(*&F1F2)F-F1F2F2*&\"#5
F2F-F2F6F1F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#pxG,(*&%\"AG\"\"\")
%\"xG\"\"#F(F(*&%\"BGF(F*F(F(%\"CGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/,,-%%diffG6$,(*&%\"AG\"\"\")%\"xG\"\"#F+F+*&%\"BGF+F-F+F+%\"CGF+-%
\"$G6$F-F.F+-F&6$F(F-F+*(F.F+F*F+F,F+!\"\"*(F.F+F0F+F-F+F8*&F.F+F1F+F8
,(*&F.F+F,F+F+*&\"#5F+F-F+F8F.F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#eqG/,.*&\"\"#\"\"\"%\"AGF)F)*(F(F)F*F)%\"xGF)F)%\"BGF)*(F(F)F*F))F,F(
F)!\"\"*(F(F)F-F)F,F)F0*&F(F)%\"CGF)F0,(*&F(F)F/F)F)*&\"#5F)F,F)F0F(F)
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The \+
constants " }{XPPEDIT 18 0 "A, B" "6$%\"AG%\"BG" }{TEXT -1 5 " and " }
{TEXT 274 1 "C" }{TEXT -1 42 " can be determined by using the followin
g " }{TEXT 0 14 "solve/identity" }{TEXT -1 8 " scheme." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "solve(i
dentity(eq,x),\{A,B,C\});\np(x)=subs(%,px);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<%/%\"AG!\"\"/%\"BG\"\"%/%\"CG\"\"!" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%\"pG6#%\"xG,&*$)F'\"\"#\"\"\"!\"\"*&\"\"%F,F'F,F,
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "This method of finding a
 particular solution, once we have identified a general form to use, i
s called the " }{TEXT 259 35 "method of undetermined coefficients" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 46 "The general solution of \+
(i) is the sum of the " }{TEXT 259 22 "complementary solution" }{TEXT 
-1 2 "  " }{XPPEDIT 18 0 "c(x) = C[1]*exp(-2*x)+C[2]*exp(x);" "6#/-%\"
cG6#%\"xG,&*&&%\"CG6#\"\"\"F--%$expG6#,$*&\"\"#F-F'F-!\"\"F-F-*&&F+6#F
3F--F/6#F'F-F-" }{TEXT -1 9 " and the " }{TEXT 259 19 "particular solu
tion" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "p(x) = 4*x-x^2;" "6#/-%\"pG6#%
\"xG,&*&\"\"%\"\"\"F'F+F+*$F'\"\"#!\"\"" }{TEXT -1 9 ", namely:" }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = C[1]*exp(-2*x)+C[
2]*exp(x)+4*x-x^2;" "6#/%\"yG,**&&%\"CG6#\"\"\"F*-%$expG6#,$*&\"\"#F*%
\"xGF*!\"\"F*F**&&F(6#F0F*-F,6#F1F*F**&\"\"%F*F1F*F**$F1F0F2" }{TEXT 
-1 16 " ------- (iii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 0 6 "dsolve" }{TEXT -1 23 " gives the same result." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
71 "de := diff(y(x),x$2)+diff(y(x),x)-2*y(x)=2*x^2-10*x+2;\ndsolve(de,
y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%
\"xG-%\"$G6$F-\"\"#\"\"\"-F(6$F*F-F2*&F1F2F*F2!\"\",(F1F2*&\"#5F2F-F2F
6*&F1F2)F-F1F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&
-%$expGF&\"\"\"%$_C2GF,F,*&-F+6#,$*&\"\"#F,F'F,!\"\"F,%$_C1GF,F,*&F'F,
,&\"\"%F4F'F,F,F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "(b) Suppo
se now that we have the initial conditions  " }{XPPEDIT 18 0 "y(0) = 3
;" "6#/-%\"yG6#\"\"!\"\"$" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y*`'`(0
) = -2;" "6#/*&%\"yG\"\"\"-%\"'G6#\"\"!F&,$\"\"#!\"\"" }{TEXT -1 2 ". \+
" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "y = 3;" "6#/%
\"yG\"\"$" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }
{TEXT -1 44 ", substituting these values in (iii) gives: " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3 = C[1]+C[2];" "6#/\"\"$,&
&%\"CG6#\"\"\"F)&F'6#\"\"#F)" }{TEXT -1 15 " ------- (iv). " }}{PARA 
0 "" 0 "" {TEXT -1 4 "Now " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "dy/dx=`` " "6#/*&%#dyG\"\"\"%#dxG!\"\"%!G" }{TEXT -1 1 
" " }{XPPEDIT 18 0 "y*`'`(x) = -2*C[1]*exp(-2*x)+C[2]*exp(x)+4-2*x;" "
6#/*&%\"yG\"\"\"-%\"'G6#%\"xGF&,**(\"\"#F&&%\"CG6#F&F&-%$expG6#,$*&F-F
&F*F&!\"\"F&F6*&&F/6#F-F&-F26#F*F&F&\"\"%F&*&F-F&F*F&F6" }{TEXT -1 2 "
. " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "dy/dx = -2;
" "6#/*&%#dyG\"\"\"%#dxG!\"\",$\"\"#F(" }{TEXT -1 6 " when " }
{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 25 " we obtain the equat
ion: " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "-2 = -2*C[1
]+C[2]+4;" "6#/,$\"\"#!\"\",(*&F%\"\"\"&%\"CG6#F)F)F&&F+6#F%F)\"\"%F)
" }{TEXT -1 1 " " }}{PARA 258 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 
256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "-6 = -2*C[1]+C[2]" "6#/,$
\"\"'!\"\",&*&\"\"#\"\"\"&%\"CG6#F*F*F&&F,6#F)F*" }{TEXT -1 14 " -----
-- (v). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
50 "Subtracting equation (v) from equation (iv) gives " }{XPPEDIT 18 
0 "9 = 3*C[1];" "6#/\"\"**&\"\"$\"\"\"&%\"CG6#F'F'" }{TEXT -1 9 " so t
hat " }{XPPEDIT 18 0 "C[1]=3" "6#/&%\"CG6#\"\"\"\"\"$" }{TEXT -1 38 ".
 We then see from equation (iv) that " }{XPPEDIT 18 0 "C[2]=0" "6#/&%
\"CG6#\"\"#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 58 "The particular solution satisfying the in
itial conditions " }{XPPEDIT 18 0 "y(0) = 3;" "6#/-%\"yG6#\"\"!\"\"$" 
}{TEXT -1 5 " and " }{XPPEDIT 18 0 "y*`'`(0) = -2;" "6#/*&%\"yG\"\"\"-
%\"'G6#\"\"!F&,$\"\"#!\"\"" }{TEXT -1 5 "  is " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "y=3*exp(-2*x)+4*x-x^2" "6#/%\"yG,(*&\"
\"$\"\"\"-%$expG6#,$*&\"\"#F(%\"xGF(!\"\"F(F(*&\"\"%F(F/F(F(*$F/F.F0" 
}{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 164 "de := diff(y(x),x$2)+diff(y(x),x)-2*y(x)=2*x^
2-10*x+2;\nic := y(0)=3,D(y)(0)=-2;\ndsolve(\{de,ic\},y(x));\ng := una
pply(rhs(%),x);\nplot(g(x),x=0..4,labels=[`x`,`y(x)`]);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#
\"\"\"-F(6$F*F-F2*&F1F2F*F2!\"\",(F1F2*&\"#5F2F-F2F6*&F1F2)F-F1F2F2" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!\"\"$/--%\"DG6#
F(F)!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&\"\"$\"
\"\"-%$expG6#,$*&\"\"#F+F'F+!\"\"F+F+*&F'F+,&\"\"%F2F'F+F+F2" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*
&\"\"$\"\"\"-%$expG6#,$*&\"\"#F/9$F/!\"\"F/F/*&F6F/,&\"\"%F7F6F/F/F7F(
F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 439 230 230 {PLOTDATA 2 "6%-%'CURVE
SG6$7W7$$\"\"!F)$\"\"$F)7$$\"3Hmmmm;')=()!#>$\"3)4u6,_#4hG!#<7$$\"3-++
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(Q&3d?F6$\"3ul51p&\\'oFF27$$\"3mmmm;6m$[#F6$\"35HkJYkKdFF27$$\"3jmmmmW
18HF6$\"3e/:>0$ycv#F27$$\"3fmmm;yYULF6$\"3%yL49E1Fw#F27$$\"3/++](GI)pP
F6$\"3G\\T-,kIxFF27$$\"3%HLL$eF>(>%F6$\"3;MFuc<d)z#F27$$\"3Qmmm\">K'*)
\\F6$\"3!Ry&\\Ux\"G&GF27$$\"3P*****\\Kd,\"eF6$\"3'yDu=iL]#HF27$$\"3-mm
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*\\,s`FF2$\"31Mj*yOvSW$F27$$\"3[mm;zM)>$GF2$\"3#)pn=y'4#=LF27$$\"3$***
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 43 ": The purel
y quadratic particular solution " }{XPPEDIT 18 0 "y=4*x-x^2" "6#/%\"yG
,&*&\"\"%\"\"\"%\"xGF(F(*$F)\"\"#!\"\"" }{TEXT -1 86 " obtained by the
 method of undetermined coefficients satisfies the initial conditions \+
" }{XPPEDIT 18 0 "y(0)=0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "y*`'`(0) = 2" "6#/*&%\"yG\"\"\"-%\"'G6#\"\"!F&\"\"#" }
{TEXT -1 59 ". This solution is of course obtained from (iii) by takin
g " }{XPPEDIT 18 0 "C[1]=0" "6#/&%\"CG6#\"\"\"\"\"!" }{TEXT -1 5 " and
 " }{XPPEDIT 18 0 "C[2]=0" "6#/&%\"CG6#\"\"#\"\"!" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
163 "de := diff(y(x),x$2)+diff(y(x),x)-2*y(x)=2*x^2-10*x+2;\nic := y(0
)=0,D(y)(0)=4;\ndsolve(\{de,ic\},y(x));\nh := unapply(rhs(%),x);\nplot
(h(x),x=0..4,labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"-F(6$F*F-F2*&F1F2
F*F2!\"\",(F1F2*&\"#5F2F-F2F6*&F1F2)F-F1F2F2" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)\"\"%" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&F'\"\"\",&\"\"%!\"\"F'F*F*
F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%
&arrowGF(,$*&9$\"\"\",&\"\"%!\"\"F.F/F/F2F(F(F(" }}{PARA 13 "" 1 "" 
{GLPLOT2D 376 252 252 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"!F)F(7$$\"3H
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u<_[EF@7$$\"3xmmm;c0T\"*F0$\"3dMnzoK$3#GF@7$$\"3#*******H,Q+5F@$\"3$H=
,b6g2+$F@7$$\"3)*******\\*3q3\"F@$\"3e(*)>EMZk;$F@7$$\"3)*******p=\\q6
F@$\"3o-REii\">J$F@7$$\"3mmm;fBIY7F@$\"3anA?m)R>V$F@7$$\"3GLLLj$[kL\"F
@$\"3O))pZD#*pfNF@7$$\"3?LLL`Q\"GT\"F@$\"3o^F;HC@bOF@7$$\"3!*****\\s]k
,:F@$\"3'*pkBmBk^PF@7$$\"39LLL`dF!e\"F@$\"3c*4gnbJQ#QF@7$$\"33++]sgam;
F@$\"3mC[By%3)))QF@7$$\"3/++]<ep[<F@$\"3(o+ey?Yo$RF@7$$\"3QLLLe/TM=F@$
\"3)Qi!p.,esRF@7$$\"3JLL$eDBJ\">F@$\"3k**HKJCX#*RF@7$$\"3immmTc-)*>F@$
\"3mr\"4>5'****RF@7$$\"3Mmm;f`@'3#F@$\"3yiP%=\"pc#*RF@7$$\"3y****\\nZ)
H;#F@$\"3=rHclfVtRF@7$$\"3YmmmJy*eC#F@$\"3dJ;QcU`RRF@7$$\"3')******R^b
JBF@$\"3G!eR\"*=r+*QF@7$$\"3f*****\\5a`T#F@$\"3G\\'fu'4[FQF@7$$\"3o***
*\\7RV'\\#F@$\"3)G%*>0P`Nv$F@7$$\"3k*****\\@fke#F@$\"3I%eT\"*elgl$F@7$
$\"3/LLL`4NnEF@$\"3]\"4&30FkaNF@7$$\"3#*******\\,s`FF@$\"3#y(R[Nf!>V$F
@7$$\"3[mm;zM)>$GF@$\"3,HPR!\\.yI$F@7$$\"3$*******pfa<HF@$\"3=fn$HR4\"
eJF@7$$\"3#HLLeg`!)*HF@$\"3+#3$))***))Q+$F@7$$\"3w****\\#G2A3$F@$\"3ul
Rq(RF)GGF@7$$\"3;LLL$)G[kJF@$\"3!poBWhzRk#F@7$$\"3#)****\\7yh]KF@$\"38
:x0(3bfV#F@7$$\"3xmmm')fdLLF@$\"33Egy(3v:A#F@7$$\"3bmmm,FT=MF@$\"3#zm]
xS0\"))>F@7$$\"3FLL$e#pa-NF@$\"3y*zoct_Bu\"F@7$$\"3!*******Rv&)zNF@$\"
3o[^I`,0/:F@7$$\"3ILLLGUYoOF@$\"3ea?x=rA;7F@7$$\"3_mmm1^rZPF@$\"3oC7Hf
!>\\X*F07$$\"34++]sI@KQF@$\"3'R4,'pD&*HkF07$$\"34++]2%)38RF@$\"3NO6*3X
F4S$F07$$\"\"%F)F(-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$%\"x
G%%y(x)G-%%VIEWG6$;F(Fcz%(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 "" }}
{PARA 0 "" 0 "" {TEXT -1 82 "If we change just one of these initial co
nditions, at least one of  the constants " }{XPPEDIT 18 0 "C[1];" "6#&
%\"CG6#\"\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "C[2];" "6#&%\"CG6#\"
\"#" }{TEXT -1 27 " will be different from 0. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "de := diff(
y(x),x$2)+diff(y(x),x)-2*y(x)=2*x^2-10*x+2;\nic := y(0)=0,D(y)(0)=21/5
;\ndsolve(\{de,ic\},y(x));\nh := unapply(rhs(%),x);\nplot(h(x),x=0..5,
labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%d
iffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"-F(6$F*F-F2*&F1F2F*F2!\"\",(F1
F2*&\"#5F2F-F2F6*&F1F2)F-F1F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)#\"#@\"\"&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&#\"\"\"\"#:F+-%$expGF&F+F+*&#F+F,F+-F
.6#,$*&\"\"#F+F'F+!\"\"F+F6*&F'F+,&\"\"%F6F'F+F+F6" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(*&#\"\"\"
\"#:F/-%$expG6#9$F/F/*&#F/F0F/-F26#,$*&\"\"#F/F4F/!\"\"F/F<*&F4F/,&\"
\"%F<F4F/F/F<F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 486 486 486 
{PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"!F)F(7$$\"3GLLL3x&)*3\"!#=$\"3#)4i
EbR)zW%F-7$$\"3umm\"H2P\"Q?F-$\"3kRcZxL/6\")F-7$$\"3MLL$eRwX5$F-$\"3?k
%QP2`0?\"!#<7$$\"33ML$3x%3yTF-$\"3[!*>`X[+p:F:7$$\"3emm\"z%4\\Y_F-$\"3
c/%eA'=l7>F:7$$\"3`LLeR-/PiF-$\"3$yC*y;z/6AF:7$$\"3]***\\il'pisF-$\"3u
f#)o%HO)*\\#F:7$$\"3>MLe*)>VB$)F-$\"3-*RH8^2sx#F:7$$\"3Y++DJbw!Q*F-$\"
3!><,#)=WC.$F:7$$\"3%ommTIOo/\"F:$\"3+$y%)*)prJF$F:7$$\"3YLL3_>jU6F:$
\"3O:t(QYPrY$F:7$$\"37++]i^Z]7F:$\"3[*pi[]Xbm$F:7$$\"33++](=h(e8F:$\"3
!*Q\"zJ)[$Q%QF:7$$\"3/++]P[6j9F:$\"3]8M*HfSh*RF:7$$\"3UL$e*[z(yb\"F:$
\"3\">7$3$*R:=TF:7$$\"3wmm;a/cq;F:$\"3F;!Hc#GXVUF:7$$\"3%ommmJ<gw\"F:$
\"3P_%3X&p8LVF:7$$\"3/+]iSj0x=F:$\"3T&zu6dR*=WF:7$$\"3gmmm\"pW`(>F:$\"
3gH3_'H;(yWF:7$$\"3K+]i!f#=$3#F:$\"38kE$40zt_%F:7$$\"3?+](=xpe=#F:$\"3
]4(pz7Oyb%F:7$$\"37nm\"H28IH#F:$\"3)RS:2bzPd%F:7$$\"3um;zpSS\"R#F:$\"3
=zInCK$[d%F:7$$\"3GLL3_?`(\\#F:$\"35qqM*yu@c%F:7$$\"3fL$e*)>pxg#F:$\"3
E)QTFpP[`%F:7$$\"33+]Pf4t.FF:$\"3<Y\\mUR:+XF:7$$\"3uLLe*Gst!GF:$\"3]#=
mQ.HBX%F:7$$\"30+++DRW9HF:$\"3yn.-2L$GR%F:7$$\"3:++DJE>>IF:$\"3%p6#Rv4
2EVF:7$$\"3F+]i!RU07$F:$\"3yKuIXE$[D%F:7$$\"3+++v=S2LKF:$\"31?dd&*Q#*p
TF:7$$\"3Jmmm\"p)=MLF:$\"3sfB&o)*H-4%F:7$$\"3B++](=]@W$F:$\"3EA=ii/u.S
F:7$$\"35L$e*[$z*RNF:$\"3*zi:ckah#RF:7$$\"3e++]iC$pk$F:$\"3@\"z@xx*oWQ
F:7$$\"3[m;H2qcZPF:$\"37*)GI(yAQx$F:7$$\"3O+]7.\"fF&QF:$\"3`u>3:/y3PF:
7$$\"3Ymm;/OgbRF:$\"3]wJYNSSdOF:7$$\"3w**\\ilAFjSF:$\"3b6fM&z40i$F:7$$
\"3yLLL$)*pp;%F:$\"3M&R8@uNbg$F:7$$\"3)RL$3xe,tUF:$\"3r*>/jk'*eh$F:7$$
\"3Cn;HdO=yVF:$\"3)QB:8y3rl$F:7$$\"3a+++D>#[Z%F:$\"3F@*GD(*yrs$F:7$$\"
3SnmT&G!e&e%F:$\"399$yc)e5_QF:7$$\"3#RLLL)Qk%o%F:$\"3rr:)[S&z5SF:7$$\"
37+]iSjE!z%F:$\"3f*>\"=\\njOUF:7$$\"3a+]P40O\"*[F:$\"3:Tq>E%pc^%F:7$$
\"\"&F)$\"3'fXw,e2U*[F:-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6
$%\"xG%%y(x)G-%%VIEWG6$;F(Fbz%(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 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 281 8 "Question" }{TEXT 
-1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 60 "(a) Find the general solution
 of the differential equation: " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "d^2*y/(d*x^2)+2" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%
\"xGF&F(!\"\"F(F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+5*y = 8*exp
(-x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"&F'%\"yGF'F'*&\"\")F'-%$ex
pG6#,$%\"xGF)F'" }{TEXT -1 14 " ------- (i). " }}{PARA 0 "" 0 "" 
{TEXT -1 74 "(b) Find the particular solution of (i) subject to the in
itial conditions " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F'" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "y*`'`(0) = -5;" "6#/*&%\"yG\"\"\"-%
\"'G6#\"\"!F&,$\"\"&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 257 "" 0 "" {TEXT 261 8 "Solution" }{TEXT 282 2 ": " }
}{PARA 0 "" 0 "" {TEXT -1 59 "(a) The solution of the corresponding ho
mogeneous equation:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "d^2*y/(d*x^2)+2" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"
F(F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+5*y = 0;" "6#/,&*&%#dyG
\"\"\"%#dxG!\"\"F'*&\"\"&F'%\"yGF'F'\"\"!" }{TEXT -1 15 " ------- (ii)
, " }}{PARA 0 "" 0 "" {TEXT -1 51 "is obtained by considering the auxi
liary equation: " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "
m^2+2*m+5 = 0;" "6#/,(*$%\"mG\"\"#\"\"\"*&F'F(F&F(F(\"\"&F(\"\"!" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 29 "Completing the square g
ives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(m+1)^2 = -
4;" "6#/*$,&%\"mG\"\"\"F'F'\"\"#,$\"\"%!\"\"" }{TEXT -1 2 ", " }}
{PARA 0 "" 0 "" {TEXT -1 9 "so that  " }{XPPEDIT 18 0 "m = -1;" "6#/%
\"mG,$\"\"\"!\"\"" }{TEXT -1 1 " " }{TEXT 283 1 "+" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "2*i" "6#*&\"\"#\"\"\"%\"iGF%" }{TEXT -1 2 ". " }}{PARA 
0 "" 0 "" {TEXT -1 43 "The general solution of (ii), which is the " }
{TEXT 259 22 "complementary solution" }{TEXT -1 12 " of (i) is: " }}
{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "c(x) = C[1]*exp(-x)*
cos*2*x+C[2]*exp(-x)*sin*2*x;" "6#/-%\"cG6#%\"xG,&*,&%\"CG6#\"\"\"F--%
$expG6#,$F'!\"\"F-%$cosGF-\"\"#F-F'F-F-*,&F+6#F4F--F/6#,$F'F2F-%$sinGF
-F4F-F'F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "deHG := diff(y(x),x$2)+2*dif
f(y(x),x)+5*y(x)=0;\ndsolve(deHG,y(x)):\nsubs(y(x)=c(x),%);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%deHGG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-
\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2*&\"\"&F2F*F2F2\"\"!" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%\"cG6#%\"xG,&*(%$_C1G\"\"\"-%$expG6#,$F'!\"\"F+-%$
sinG6#,$*&\"\"#F+F'F+F+F+F+*(%$_C2GF+F,F+-%$cosGF3F+F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 104 "It is likely that we could find a part
icular solution of (i) having the form of an exponential function " }
{XPPEDIT 18 0 "p(x) = A*exp(-x);" "6#/-%\"pG6#%\"xG*&%\"AG\"\"\"-%$exp
G6#,$F'!\"\"F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " 
}{TEXT 284 1 "y" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "p(x) = A*exp(-x);" 
"6#/-%\"pG6#%\"xG*&%\"AG\"\"\"-%$expG6#,$F'!\"\"F*" }{TEXT -1 10 ", we
 have " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "dy/dx = p*
`'`(x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%\"pGF&-%\"'G6#%\"xGF&" }
{XPPEDIT 18 0 "`` = -A*exp(-x);" "6#/%!G,$*&%\"AG\"\"\"-%$expG6#,$%\"x
G!\"\"F(F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = p*`''
`(x);" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"*&%\"pGF(-%#''
G6#F+F(" }{XPPEDIT 18 0 "`` = A*exp(-x);" "6#/%!G*&%\"AG\"\"\"-%$expG6
#,$%\"xG!\"\"F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 44 "Substi
tuting the last three expressions for " }{XPPEDIT 18 0 "d^2*y/(d*x^2)
" "6#*(%\"dG\"\"#%\"yG\"\"\"*&F$F'*$%\"xGF%F'!\"\"" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 5 " and " 
}{TEXT 286 1 "y" }{TEXT -1 27 " in (i) gives the equation " }}{PARA 
256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "A*exp(-x)+2*(-A*exp(-x))+5
*A*exp(-x) = 8*exp(-x);" "6#/,(*&%\"AG\"\"\"-%$expG6#,$%\"xG!\"\"F'F'*
&\"\"#F',$*&F&F'-F)6#,$F,F-F'F-F'F'*(\"\"&F'F&F'-F)6#,$F,F-F'F'*&\"\")
F'-F)6#,$F,F-F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }
}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "4*A*exp(-x)=8*exp(-
x)" "6#/*(\"\"%\"\"\"%\"AGF&-%$expG6#,$%\"xG!\"\"F&*&\"\")F&-F)6#,$F,F
-F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 94 "The left and righ
t sides of this last equation are identically equal (equal for all val
ues of " }{TEXT 285 1 "x" }{TEXT -1 16 ") provided that " }{XPPEDIT 
18 0 "A=2" "6#/%\"AG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 18 "Hence (i) has the " }{TEXT 259 19 "particular solution" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "p(x)=2*exp(-x)" "6#/-%\"pG6#%\"xG*&\"\"#\"\"
\"-%$expG6#,$F'!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 46 "The general solution of (i) is the s
um of the " }{TEXT 259 23 "complementary solution " }}{PARA 256 "" 0 "
" {TEXT -1 2 "  " }{XPPEDIT 18 0 "c(x) = C[1]*exp(-x)*cos*2*x+C[2]*exp
(-x)*sin*2*x;" "6#/-%\"cG6#%\"xG,&*,&%\"CG6#\"\"\"F--%$expG6#,$F'!\"\"
F-%$cosGF-\"\"#F-F'F-F-*,&F+6#F4F--F/6#,$F'F2F-%$sinGF-F4F-F'F-F-" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "and the " }{TEXT 259 19 "
particular solution" }{TEXT -1 1 " " }{XPPEDIT 18 0 "p(x) = 2*exp(-x)
" "6#/-%\"pG6#%\"xG*&\"\"#\"\"\"-%$expG6#,$F'!\"\"F*" }{TEXT -1 2 ". \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The g
eneral solution of (i) is therefore: " }}{PARA 256 "" 0 "" {TEXT -1 2 
"  " }{XPPEDIT 18 0 "y = C[1]*exp(-x)*cos*2*x+C[2]*exp(-x)*sin*2*x+2*e
xp(-x);" "6#/%\"yG,(*,&%\"CG6#\"\"\"F*-%$expG6#,$%\"xG!\"\"F*%$cosGF*
\"\"#F*F/F*F**,&F(6#F2F*-F,6#,$F/F0F*%$sinGF*F2F*F/F*F**&F2F*-F,6#,$F/
F0F*F*" }{TEXT -1 16 " ------- (iii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "de := diff(y(x),x$2)+2*
diff(y(x),x)+5*y(x)=8*exp(-x);\nunassign('A'):\npx := A*exp(-x);\nsubs
(y(x)=px,de);\nsimplify(%);\nsolve(identity(%,x),A);\np(x)=subs(A=%,px
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG
-%\"$G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2*&\"\"&F2F*F2F2,$*&\"\")F2-%$e
xpG6#,$F-!\"\"F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#pxG*&%\"AG\"
\"\"-%$expG6#,$%\"xG!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%%d
iffG6$*&%\"AG\"\"\"-%$expG6#,$%\"xG!\"\"F*-%\"$G6$F/\"\"#F**&F4F*-F&6$
F(F/F*F**(\"\"&F*F)F*F+F*F*,$*&\"\")F*F+F*F*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,$*(\"\"%\"\"\"%\"AGF'-%$expG6#,$%\"xG!\"\"F'F',$*&\"
\")F'F)F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/-%\"pG6#%\"xG,$*&\"\"#\"\"\"-%$expG6#,$F'!\"\"F+F+
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "We c
an obtain the general solution using " }{TEXT 0 6 "dsolve" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 70 "de := diff(y(x),x$2)+2*diff(y(x),x)+5*y(x)=8*exp(-x);
\ndsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diff
G6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2*&\"\"&F2F*F2F
2,$*&\"\")F2-%$expG6#,$F-!\"\"F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/-%\"yG6#%\"xG,(*(-%$expG6#,$F'!\"\"\"\"\"-%$sinG6#,$*&\"\"#F/F'F/F/F/
%$_C2GF/F/*(F*F/-%$cosGF2F/%$_C1GF/F/*&F5F/F*F/F/" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 52 "(b) Suppose now that we have the initial condit
ions " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 5 " a
nd " }{XPPEDIT 18 0 "y*`'`(0) = -5;" "6#/*&%\"yG\"\"\"-%\"'G6#\"\"!F&,
$\"\"&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }
{XPPEDIT 18 0 "y = 0;" "6#/%\"yG\"\"!" }{TEXT -1 6 " when " }{XPPEDIT 
18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 43 ", substituting these values i
n (iii) gives " }{XPPEDIT 18 0 "0 = C[1]+2;" "6#/\"\"!,&&%\"CG6#\"\"\"
F)\"\"#F)" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "C[1]=-2" "6#/&%\"
CG6#\"\"\",$\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
42 "Equation (iii) can be written in the form " }}{PARA 256 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "y=exp(-x)*(C[1]*cos*2*x+C[2]*sin*2*x+2
)" "6#/%\"yG*&-%$expG6#,$%\"xG!\"\"\"\"\",(**&%\"CG6#F,F,%$cosGF,\"\"#
F,F*F,F,**&F06#F3F,%$sinGF,F3F,F*F,F,F3F,F," }{TEXT -1 2 ", " }}{PARA 
0 "" 0 "" {TEXT -1 55 "so that, using the rule for differentiating a p
roduct: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=y*`
'`(x)" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%\"yGF&-%\"'G6#%\"xGF&" }{XPPEDIT 
18 0 "`` = -exp(-x)*(C[1]*cos*2*x+C[2]*sin*2*x+2)+exp(-x)*(-2*C[1]*sin
*2*x+2*C[2]*cos*2*x);" "6#/%!G,&*&-%$expG6#,$%\"xG!\"\"\"\"\",(**&%\"C
G6#F-F-%$cosGF-\"\"#F-F+F-F-**&F16#F4F-%$sinGF-F4F-F+F-F-F4F-F-F,*&-F(
6#,$F+F,F-,&*,F4F-&F16#F-F-F8F-F4F-F+F-F,*,F4F-&F16#F4F-F3F-F4F-F+F-F-
F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }
{XPPEDIT 18 0 "dy/dx=-5" "6#/*&%#dyG\"\"\"%#dxG!\"\",$\"\"&F(" }{TEXT 
-1 7 ", when " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 25 " we \+
obtain the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "-5=-(C[1]+2)+2*C[2]" "6#/,$\"\"&!\"\",&,&&%\"CG6#\"\"\"F,\"\"#F,
F&*&F-F,&F*6#F-F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Si
nce " }{XPPEDIT 18 0 "C[1]=-2" "6#/&%\"CG6#\"\"\",$\"\"#!\"\"" }{TEXT 
-1 14 ", we see that " }{XPPEDIT 18 0 "C[2]=-5/2" "6#/&%\"CG6#\"\"#,$*
&\"\"&\"\"\"F'!\"\"F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 58 "The particular solution satisfying th
e initial conditions " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F'
" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y*`'`(0) = -5;" "6#/*&%\"yG\"\"
\"-%\"'G6#\"\"!F&,$\"\"&!\"\"" }{TEXT -1 6 "  is: " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=-2*exp(-x)*cos*2*x-5/2" "6#/%\"yG,&
*,\"\"#\"\"\"-%$expG6#,$%\"xG!\"\"F(%$cosGF(F'F(F-F(F.*&\"\"&F(F'F.F.
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-x)*sin*2*x+2*exp(-x)" "6#,&**-%
$expG6#,$%\"xG!\"\"\"\"\"%$sinGF+\"\"#F+F)F+F+*&F-F+-F&6#,$F)F*F+F+" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 163 "de := diff(y(x),x$2)+2*diff(y(x),x)+5*y(x)=8*ex
p(-x);\nic := y(0)=0,D(y)(0)=-5;\ndsolve(\{de,ic\},y(x));\ng := unappl
y(rhs(%),x):\nplot(g(x),x=0..4,labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"
\"*&F1F2-F(6$F*F-F2F2*&\"\"&F2F*F2F2,$*&\"\")F2-%$expG6#,$F-!\"\"F2F2
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#
F(F)!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&#\"\"&\"
\"#\"\"\"*&-%$expG6#,$F'!\"\"F--%$sinG6#,$*&F,F-F'F-F-F-F-F3*(F,F-F/F-
-%$cosGF6F-F3*&F,F-F/F-F-" }}{PARA 13 "" 1 "" {GLPLOT2D 497 305 305 
{PLOTDATA 2 "6%-%'CURVESG6$7eo7$$\"\"!F)F(7$$\"3emmm;arz@!#>$!3;*>3rGH
u/\"!#=7$$\"39LLLL3VfVF-$!3<R\\zx6O6?F07$$\"3s******\\i9RlF-$!3/()yaq;
!R*GF07$$\"3Hmmmm;')=()F-$!33-sRt\"Rsp$F07$$\"3-++]7z>^7F0$!3Q%[3D!H78
\\F07$$\"3RLLLe'40j\"F0$!3k:r!p[C%3fF07$$\"3/++](Q&3d?F0$!3%\\Dn='3C!y
'F07$$\"3mmmm;6m$[#F0$!3Nc'\\fT(=3uF07$$\"3Ommm\"zi$)p#F0$!36gY/Lf.QwF
07$$\"3jmmmmW18HF0$!3,'[=jETM\"yF07$$\"3*omm;9mx7$F0$!3P$H0FP\")o$zF07
$$\"3fmmm;yYULF0$!3)\\GZ)Q[$3,)F07$$\"3C+]PM%3$\\MF0$!3uAN#Gej*H!)F07$
$\"3KLL3_!\\hb$F0$!3Cb#fLzax.)F07$$\"3Sm;zp'*)Hm$F0$!3c%**o]w6X.)F07$$
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\"3K+++DI(yv)F0$!3Y.8Yc:e^TF-7$$\"3xmmm;c0T\"*F0$\"3m4DY'e%QpOF-7$$\"3
+LLLe%GCd*F0$\"3hyN4PR9H7F07$$\"3#*******H,Q+5!#<$\"3ugb/>L%Q1#F07$$\"
3&*******RXpV5F_u$\"3cU?:UzwmGF07$$\"3)*******\\*3q3\"F_u$\"39HGPuh!zi
$F07$$\"3)*******p=\\q6F_u$\"3'>dgB5,v&\\F07$$\"3mmm;fBIY7F_u$\"3U**3G
^YR))fF07$$\"3GLLLj$[kL\"F_u$\"3u#*z)[()en(pF07$$\"3?LLL`Q\"GT\"F_u$\"
3`b%3#=K*fg(F07$$\"3!*****\\s]k,:F_u$\"3'*o2$[7H-5)F07$$\"3_mm\"HTg4a
\"F_u$\"3k!z#\\z?2S#)F07$$\"39LLL`dF!e\"F_u$\"3AZ2u3deL$)F07$$\"3^mm\"
H\"4TB;F_u$\"3!*QBusD1&Q)F07$$\"33++]sgam;F_u$\"3oK=;?^q&Q)F07$$\"3&**
****\\%4i2<F_u$\"3\\WP#=(p\"=M)F07$$\"3/++]<ep[<F_u$\"3=FVY5JNd#)F07$$
\"3qmm\"z8`:z\"F_u$\"3r[\"f/j[\"H\")F07$$\"3QLLLe/TM=F_u$\"3'4&z'H<PM'
zF07$$\"3JLL$eDBJ\">F_u$\"3o.=p30EtvF07$$\"3immmTc-)*>F_u$\"3W%\\mIu0(
\\qF07$$\"3Mmm;f`@'3#F_u$\"33DM'y)z+AkF07$$\"3y****\\nZ)H;#F_u$\"3BPc?
_*y)GeF07$$\"3YmmmJy*eC#F_u$\"3w8UD)y)[h^F07$$\"3')******R^bJBF_u$\"3Q
5SA<7EkWF07$$\"3f*****\\5a`T#F_u$\"3+.J#H&=e$z$F07$$\"3o****\\7RV'\\#F
_u$\"3%[#R#\\dr/<$F07$$\"3k*****\\@fke#F_u$\"3[r=J1:hADF07$$\"3/LLL`4N
nEF_u$\"3G;*zp`&o*)>F07$$\"3#*******\\,s`FF_u$\"3_/'o*QgTz9F07$$\"3[mm
;zM)>$GF_u$\"3SJp?X$zK2\"F07$$\"3$*******pfa<HF_u$\"3N6Q!)esbBpF-7$$\"
3#HLLeg`!)*HF_u$\"3kCdkP.[RRF-7$$\"3w****\\#G2A3$F_u$\"3G8n/T#fIU\"F-7
$$\"3;LLL$)G[kJF_u$!3'Q[(4)*GwVZ!#?7$$\"3#)****\\7yh]KF_u$!3*Q7()RIk@
\">F-7$$\"3xmmm')fdLLF_u$!3*y'*\\X#Ho?GF-7$$\"3bmmm,FT=MF_u$!3'y)*Q4R!
)yK$F-7$$\"3FLL$e#pa-NF_u$!3#ptz7G;LZ$F-7$$\"3!*******Rv&)zNF_u$!3U_Ix
mS@[LF-7$$\"3ILLLGUYoOF_u$!3=sF2j&3X'HF-7$$\"3_mmm1^rZPF_u$!3OzI&[>i\"
eCF-7$$\"34++]sI@KQF_u$!37?Xh!z(Q-=F-7$$\"34++]2%)38RF_u$!3Lt'['36436F
-7$$\"\"%F)$!3La*>$45pSLF]_l-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLAB
ELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F\\bl%(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 9 "Example 3" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 287 8 "Question" }{TEXT 
-1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 59 "(a) Find the general solution
 of the differential equation:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "d^2*y/(d*x^2)+2;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%
\"xGF&F(!\"\"F(F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+y = 25*sin*
2*x;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'%\"yGF'**\"#DF'%$sinGF'\"\"#F'%\"
xGF'" }{TEXT -1 15 "  ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 74 "(b) Find the particular solution of (i) \+
subject to the initial conditions " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%
\"yG6#\"\"!F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y*`'`(0) = 0;" "6#/
*&%\"yG\"\"\"-%\"'G6#\"\"!F&F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 261 8 "Solution" }{TEXT 288 2 
": " }}{PARA 0 "" 0 "" {TEXT -1 59 "(a) The solution of the correspond
ing homogeneous equation:" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "d^2*y/(d*x^2)+2;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%
\"xGF&F(!\"\"F(F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+y = 0;" "6#
/,&*&%#dyG\"\"\"%#dxG!\"\"F'%\"yGF'\"\"!" }{TEXT -1 15 " ------- (ii),
 " }}{PARA 0 "" 0 "" {TEXT -1 50 "is obtained by considering the auxil
iary equation:" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "m^
2+2*m+1 = 0;" "6#/,(*$%\"mG\"\"#\"\"\"*&F'F(F&F(F(F(F(\"\"!" }{TEXT 
-1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "This equation is equivalent t
o " }{XPPEDIT 18 0 "(m+1)^2 = 0;" "6#/*$,&%\"mG\"\"\"F'F'\"\"#\"\"!" }
{TEXT -1 29 ", and so has the single root " }{XPPEDIT 18 0 "m = -1;" "
6#/%\"mG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 43 "The general solution of (ii), which i
s the " }{TEXT 259 22 "complementary solution" }{TEXT -1 12 " of (i) i
s: " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "c(x) = C[1]*e
xp(-x)+C[2]*x*exp(-x);" "6#/-%\"cG6#%\"xG,&*&&%\"CG6#\"\"\"F--%$expG6#
,$F'!\"\"F-F-*(&F+6#\"\"#F-F'F--F/6#,$F'F2F-F-" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
83 "deHG := diff(y(x),x$2)+2*diff(y(x),x)+y(x)=0;\ndsolve(deHG,y(x)):
\nsubs(y(x)=c(x),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%deHGG/,(-%%
diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2F*F2\"\"!
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"cG6#%\"xG,&*&%$_C1G\"\"\"-%$e
xpG6#,$F'!\"\"F+F+*(%$_C2GF+F,F+F'F+F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 48 "A trial particular solution for (i) of the form " }
{XPPEDIT 18 0 "p(x) = A*cos*2*x+B*sin*2*x;" "6#/-%\"pG6#%\"xG,&**%\"AG
\"\"\"%$cosGF+\"\"#F+F'F+F+**%\"BGF+%$sinGF+F-F+F'F+F+" }{TEXT -1 18 "
  is appropriate. " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{TEXT 289 
1 "y" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "p(x) = A*cos*2*x+B*sin*2*x;" "
6#/-%\"pG6#%\"xG,&**%\"AG\"\"\"%$cosGF+\"\"#F+F'F+F+**%\"BGF+%$sinGF+F
-F+F'F+F+" }{TEXT -1 10 ", we have:" }}{PARA 256 "" 0 "" {TEXT -1 2 " \+
 " }{XPPEDIT 18 0 "dy/dx = p*`'`(x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%\"
pGF&-%\"'G6#%\"xGF&" }{XPPEDIT 18 0 "`` = -2*A*sin*2*x+2*B*cos*2*x;" "
6#/%!G,&*,\"\"#\"\"\"%\"AGF(%$sinGF(F'F(%\"xGF(!\"\"*,F'F(%\"BGF(%$cos
GF(F'F(F+F(F(" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}
{PARA 256 "" 0 "" {TEXT -1 4 "    " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = p*
`''`(x);" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"*&%\"pGF(-%
#''G6#F+F(" }{XPPEDIT 18 0 "`` = -4*A*cos*2*x-4*B*sin*2*x;" "6#/%!G,&*
,\"\"%\"\"\"%\"AGF(%$cosGF(\"\"#F(%\"xGF(!\"\"*,F'F(%\"BGF(%$sinGF(F+F
(F,F(F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "Substituting \+
the last three expressions for " }{XPPEDIT 18 0 "d^2*y/(d*x^2)" "6#*(%
\"dG\"\"#%\"yG\"\"\"*&F$F'*$%\"xGF%F'!\"\"" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 5 " and " 
}{TEXT 290 1 "y" }{TEXT -1 28 " in (i) gives the equation: " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(-4*A*cos*2*x-4*B*sin*2*x
)+2*(-2*A*sin*2*x+2*B*cos*2*x)+A*cos*2*x+B*sin*2*x = 25*sin*2*x;" "6#/
,*-%!G6#,&*,\"\"%\"\"\"%\"AGF+%$cosGF+\"\"#F+%\"xGF+!\"\"*,F*F+%\"BGF+
%$sinGF+F.F+F/F+F0F+*&F.F+,&*,F.F+F,F+F3F+F.F+F/F+F0*,F.F+F2F+F-F+F.F+
F/F+F+F+F+**F,F+F-F+F.F+F/F+F+**F2F+F3F+F.F+F/F+F+**\"#DF+F3F+F.F+F/F+
" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 10 "that is,  " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(-3*A+4*B)*cos*2*x+(-4*A-3*
B)*sin*2*x = 25*sin*2*x;" "6#/,&**,&*&\"\"$\"\"\"%\"AGF)!\"\"*&\"\"%F)
%\"BGF)F)F)%$cosGF)\"\"#F)%\"xGF)F)**,&*&F-F)F*F)F+*&F(F)F.F)F+F)%$sin
GF)F0F)F1F)F)**\"#DF)F6F)F0F)F1F)" }{TEXT -1 16 " ------- (iii). " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "We wish t
o find the values of the undetermined coefficients " }{TEXT 291 1 "A" 
}{TEXT -1 5 " and " }{TEXT 292 1 "B" }{TEXT -1 31 " which make (iii) a
n identity. " }}{PARA 0 "" 0 "" {TEXT -1 37 "We can can obtain equatio
ns relating " }{TEXT 293 1 "A" }{TEXT -1 5 " and " }{TEXT 294 1 "B" }
{TEXT -1 34 " by comparing the coefficients of " }{XPPEDIT 18 0 "cos*2
*x" "6#*(%$cosG\"\"\"\"\"#F%%\"xGF%" }{TEXT -1 5 " and " }{XPPEDIT 18 
0 "sin*2*x" "6#*(%$sinG\"\"\"\"\"#F%%\"xGF%" }{TEXT -1 39 " on the lef
t and right sides of (iii). " }}{PARA 0 "" 0 "" {TEXT -1 29 "Equating \+
the coefficients of " }{XPPEDIT 18 0 "cos*2*x" "6#*(%$cosG\"\"\"\"\"#F
%%\"xGF%" }{TEXT -1 9 "  gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "-3*A+4*B = 0" "6#/,&*&\"\"$\"\"\"%\"AGF'!\"\"*&\"\"%F'%
\"BGF'F'\"\"!" }{TEXT -1 15 " ------- (iv). " }}{PARA 0 "" 0 "" {TEXT 
-1 47 "This equation is also obtained by substituting " }{XPPEDIT 18 
0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 11 " in (iii). " }}{PARA 0 "" 0 "" 
{TEXT -1 29 "Equating the coefficients of " }{XPPEDIT 18 0 "sin*2*x;" 
"6#*(%$sinG\"\"\"\"\"#F%%\"xGF%" }{TEXT -1 9 "  gives: " }}{PARA 256 "
" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "-4*A-3*B = 25" "6#/,&*&\"\"%\"
\"\"%\"AGF'!\"\"*&\"\"$F'%\"BGF'F)\"#D" }{TEXT -1 14 " ------- (v). " 
}}{PARA 0 "" 0 "" {TEXT -1 47 "This equation is also obtained by subst
ituting " }{XPPEDIT 18 0 "x = Pi/4;" "6#/%\"xG*&%#PiG\"\"\"\"\"%!\"\"
" }{TEXT -1 22 " in (iii). This makes " }{XPPEDIT 18 0 "cos*2*x" "6#*(
%$cosG\"\"\"\"\"#F%%\"xGF%" }{TEXT -1 16 " equal to 0 and " }{XPPEDIT 
18 0 "sin*2*x" "6#*(%$sinG\"\"\"\"\"#F%%\"xGF%" }{TEXT -1 13 " equal t
o 1. " }}{PARA 0 "" 0 "" {TEXT -1 58 "The two equations (iv) and (v) c
an readily be solved  for " }{TEXT 295 1 "A" }{TEXT -1 5 " and " }
{TEXT 296 1 "B" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "Multip
lying (iv) by 3 and (v) by 4 gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{XPPEDIT 18 0 "PIECEWISE([-9*A+12*B=0,``],[-16*A-12*B=100,``])" "6
#-%*PIECEWISEG6$7$/,&*&\"\"*\"\"\"%\"AGF+!\"\"*&\"#7F+%\"BGF+F+\"\"!%!
G7$/,&*&\"#;F+F,F+F-*&F/F+F0F+F-\"$+\"F2" }{TEXT -1 2 ". " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Adding these two e
quations gives " }{XPPEDIT 18 0 "-25*A=100" "6#/,$*&\"#D\"\"\"%\"AGF'!
\"\"\"$+\"" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "A=-4" "6#/%\"AG,$
\"\"%!\"\"" }{TEXT -1 51 ". Then substituting this value for A in (iv)
 gives " }{XPPEDIT 18 0 "12+4*B=0" "6#/,&\"#7\"\"\"*&\"\"%F&%\"BGF&F&
\"\"!" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "B=-3" "6#/%\"BG,$\"\"$!
\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 
"" {TEXT -1 19 "Thus we obtain the " }{TEXT 259 19 "particular solutio
n" }{TEXT -1 1 " " }{XPPEDIT 18 0 "p(x) = -4*cos*2*x-3*sin*2*x;" "6#/-
%\"pG6#%\"xG,&**\"\"%\"\"\"%$cosGF+\"\"#F+F'F+!\"\"**\"\"$F+%$sinGF+F-
F+F'F+F." }{TEXT -1 11 "  for (i). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 46 "The general solution of (i) is the sum o
f the " }{TEXT 259 22 "complementary solution" }{TEXT -1 2 "  " }
{XPPEDIT 18 0 "c(x) = C[1]*exp(-x)+C[2]*x*exp(-x)" "6#/-%\"cG6#%\"xG,&
*&&%\"CG6#\"\"\"F--%$expG6#,$F'!\"\"F-F-*(&F+6#\"\"#F-F'F--F/6#,$F'F2F
-F-" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 8 "and the " }{TEXT 
259 19 "particular solution" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "p(x) = -
4*cos(2*x)-3*sin(2*x);" "6#/-%\"pG6#%\"xG,&*&\"\"%\"\"\"-%$cosG6#*&\"
\"#F+F'F+F+!\"\"*&\"\"$F+-%$sinG6#*&F0F+F'F+F+F1" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "The gener
al solution of (i) is therefore:  " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "y = C[1]*exp(-x)+C[2]*x*exp(-x)-4*cos*2*x-3*sin*2*x;
" "6#/%\"yG,**&&%\"CG6#\"\"\"F*-%$expG6#,$%\"xG!\"\"F*F**(&F(6#\"\"#F*
F/F*-F,6#,$F/F0F*F***\"\"%F*%$cosGF*F4F*F/F*F0**\"\"$F*%$sinGF*F4F*F/F
*F0" }{TEXT -1 15 " ------- (vi). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The const
ants " }{TEXT 298 1 "A" }{TEXT -1 5 " and " }{TEXT 299 1 "B" }{TEXT 
-1 28 " can be determined by using " }{TEXT 297 0 "" }{TEXT 0 14 "solv
e/identity" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "de := diff(y(x),x$2)+2*diff(y(x),x
)+y(x)=25*sin(2*x);\nunassign('A','B'):\npx := A*cos(2*x)+B*sin(2*x);
\nsubs(y(x)=px,de);\neq := simplify(%);\nsolve(identity(eq,x),\{A,B\})
;\np(x)=subs(%,px);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%dif
fG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2F*F2,$*&\"#DF
2-%$sinG6#,$*&F1F2F-F2F2F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#pxG
,&*&%\"AG\"\"\"-%$cosG6#,$*&\"\"#F(%\"xGF(F(F(F(*&%\"BGF(-%$sinGF+F(F(
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*-%%diffG6$,&*&%\"AG\"\"\"-%$cos
G6#,$*&\"\"#F+%\"xGF+F+F+F+*&%\"BGF+-%$sinGF.F+F+-%\"$G6$F2F1F+*&F1F+-
F&6$F(F2F+F+F)F+F3F+,$*&\"#DF+F5F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#eqG/,**(\"\"$\"\"\"%\"AGF)-%$cosG6#,$*&\"\"#F)%\"xGF)F)F)!\"\"*(F
(F)%\"BGF)-%$sinGF-F)F2*(\"\"%F)F*F)F5F)F2*(F8F)F4F)F+F)F),$*&\"#DF)F5
F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"AG!\"%/%\"BG!\"$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG,&*&\"\"%\"\"\"-%$cosG6#
,$*&\"\"#F+F'F+F+F+!\"\"*&\"\"$F+-%$sinGF.F+F2" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 35 "We can check the general solution: " }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(x) = C[1]*exp(-x)+C[2]*x*exp(-
x)-4*cos(2*x)-3*sin(2*x);" "6#/-%\"yG6#%\"xG,**&&%\"CG6#\"\"\"F--%$exp
G6#,$F'!\"\"F-F-*(&F+6#\"\"#F-F'F--F/6#,$F'F2F-F-*&\"\"%F--%$cosG6#*&F
6F-F'F-F-F2*&\"\"$F--%$sinG6#*&F6F-F'F-F-F2" }{TEXT -1 1 " " }}{PARA 
0 "" 0 "" {TEXT -1 68 " by substituting it into the original non-homog
eneous equation (i). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 131 "de := diff(y(x),x$2)+2*diff(y(x),x)+y(x)
=25*sin(2*x);\nsubs(y(x)=C[1]*exp(-x)+C[2]*x*exp(-x)-4*cos(2*x)-3*sin(
2*x),de);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%
%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2F*F2,$-%$
sinG6#,$F-F1\"#D" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,.-%%diffG6$,**&&
%\"CG6#\"\"\"F--%$expG6#,$%\"xG!\"\"F-F-*(&F+6#\"\"#F-F2F-F.F-F-*&\"\"
$F--%$sinG6#,$F2F7F-F3*&\"\"%F--%$cosGF<F-F3-%\"$G6$F2F7F-*&F7F--F&6$F
(F2F-F-F)F-F4F-*&F9F-F:F-F3*&F?F-F@F-F3,$F:\"#D" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,$-%$sinG6#,$%\"xG\"\"#\"#DF$" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "We can obtain the general
 solution using " }{TEXT 0 6 "dsolve" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "de := diff
(y(x),x$2)+2*diff(y(x),x)+y(x)=25*sin(2*x);\ndsolve(de,y(x));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G
6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2F*F2,$-%$sinG6#,$F-F1\"#D" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,**&-%$expG6#,$F'!\"\"\"\"\"%$
_C2GF/F/*(F*F/F'F/%$_C1GF/F/*&\"\"%F/-%$cosG6#,$F'\"\"#F/F.*&\"\"$F/-%
$sinGF7F/F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 52 "(b) Suppose now that we have the initial conditions " }
{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "y*`'`(0) = 0;" "6#/*&%\"yG\"\"\"-%\"'G6#\"\"!F&F*" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "
y = 0;" "6#/%\"yG\"\"!" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "x=0" "6#/
%\"xG\"\"!" }{TEXT -1 42 ", substituting these values in (vi) gives " 
}{XPPEDIT 18 0 "0 = C[1]-4;" "6#/\"\"!,&&%\"CG6#\"\"\"F)\"\"%!\"\"" }
{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "C[1] = 4;" "6#/&%\"CG6#\"\"\"
\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 41 "Equation (vi) c
an be written in the form " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "y = exp(-x)*(C[1]+C[2]*x)-4*cos*2*x-3*sin*2*x;" "6#/%\"
yG,(*&-%$expG6#,$%\"xG!\"\"\"\"\",&&%\"CG6#F-F-*&&F06#\"\"#F-F+F-F-F-F
-**\"\"%F-%$cosGF-F5F-F+F-F,**\"\"$F-%$sinGF-F5F-F+F-F," }{TEXT -1 2 "
, " }}{PARA 0 "" 0 "" {TEXT -1 9 "so that: " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=y*`'`(x)" "6#/*&%#dyG\"\"\"%#dxG!
\"\"*&%\"yGF&-%\"'G6#%\"xGF&" }{XPPEDIT 18 0 "`` = -exp(-x)*(C[1]+C[2]
*x)+exp(-x)*C[2]+8*sin*2*x-6*cos*2*x;" "6#/%!G,**&-%$expG6#,$%\"xG!\"
\"\"\"\",&&%\"CG6#F-F-*&&F06#\"\"#F-F+F-F-F-F,*&-F(6#,$F+F,F-&F06#F5F-
F-**\"\")F-%$sinGF-F5F-F+F-F-**\"\"'F-%$cosGF-F5F-F+F-F," }{TEXT -1 2 
". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "dy/dx = 0;
" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 7 ", when " }{XPPEDIT 
18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 25 " we obtain the equation: " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "0 = -C[1]+C[2]-6;" "6
#/\"\"!,(&%\"CG6#\"\"\"!\"\"&F'6#\"\"#F)\"\"'F*" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "C[1] = 4;" "6#/&%
\"CG6#\"\"\"\"\"%" }{TEXT -1 14 ", we see that " }{XPPEDIT 18 0 "C[2] \+
= 10;" "6#/&%\"CG6#\"\"#\"#5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "The particular solution s
atisfying the initial conditions " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"
yG6#\"\"!F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y*`'`(0) = -5;" "6#/*
&%\"yG\"\"\"-%\"'G6#\"\"!F&,$\"\"&!\"\"" }{TEXT -1 6 "  is: " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=4*exp(-x)+10*x*exp(-x)-4*
cos*2*x-3*sin*2*x" "6#/%\"yG,**&\"\"%\"\"\"-%$expG6#,$%\"xG!\"\"F(F(*(
\"#5F(F-F(-F*6#,$F-F.F(F(**F'F(%$cosGF(\"\"#F(F-F(F.**\"\"$F(%$sinGF(F
6F(F-F(F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "de := diff(y(x),x$2)+2*diff(y(x),x
)+y(x)=25*sin(2*x);\nic := y(0)=0,D(y)(0)=0;\ndsolve(\{de,ic\},y(x));
\ng := unapply(rhs(%),x);\nplot(g(x),x=0..12.2,labels=[`x`,`y(x)`]);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"
$G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2F*F2,$*&\"#DF2-%$sinG6#,$*&F1F2F-F
2F2F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--
%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,**&\"
\"%\"\"\"-%$expG6#,$F'!\"\"F+F+*(\"#5F+F,F+F'F+F+*&F*F+-%$cosG6#,$*&\"
\"#F+F'F+F+F+F0*&\"\"$F+-%$sinGF6F+F0" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,**&\"\"%\"\"\"-%$expG6#,$
9$!\"\"F/F/*(\"#5F/F0F/F4F/F/*&F.F/-%$cosG6#,$*&\"\"#F/F4F/F/F/F5*&\"
\"$F/-%$sinGF;F/F5F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 425 289 289 
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\"3im\"zB\"oFo5F_`m$\"37)))fBgz\")[\"FP7$$\"3?$eR5om+3\"F_`m$\"3mZenn
\"\\@c#FP7$$\"3%*****p\\l&=4\"F_`m$\"3=*R\"=1NZ%\\$FP7$$\"3LL3np!p`5\"
F_`m$\"35%3S/1PFK%FP7$$\"3rm;k*e\"))=6F_`m$\"3&ex746kt$[FP7$$\"3,D\"y$
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V6F_`m$\"3!pir2ouO([FP7$$\"3m;HKZ!Rf:\"F_`m$\"3kLPV!yRlU%FP7$$\"31+D6(
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-+vGk>\\$>\"F_`m$\"3y(Qx0Q>vk\"FP7$$\"3*)\\P9#)fu17F_`m$\"3P[E*ePc0`$F
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F(F(-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F($\"$A\"F[hm%(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 42 "Table of sample trial p
articular solutions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 77 "The following table gives examples of app
ropriate trial particular solutions " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6
#%\"xG" }{TEXT -1 73 " for a 2nd order linear differential equation wi
th constant coefficients:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "a;" "6#%
\"aG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+b;" "6#,&*(%\"dG\"
\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(%\"bGF(" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "dy/dx+c*y = f(x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&%\"
cGF'%\"yGF'F'-%\"fG6#%\"xG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 32 "with a given \"forcing function\" " }{XPPEDIT 18 0 "f(x)" "6#-%
\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "matrix([[f(x), `                 Form of trial particul
ar solution`*p(x)*`           `], [any*constant, A], [8*x-5, A*x+B], [
x^2-7*x-1, A*x^2+B*x+C], [2*x^3-x+9, A*x^3+B*x^2+C*x+D], [3*exp(4*x), \+
A*exp(4*x)], [5*sin*3*x, A*cos*3*x+B*sin*3*x], [-4*cos*2*x, A*cos*2*x+
B*sin*2*x], [3*x*exp(-x), (A*x+B)*exp(-x)], [x^2*exp(x/2), (A*x^2+B*x+
C)*exp(x/2)], [exp(2*x)*sin*x, A*exp(2*x)*cos*x+B*exp(2*x)*sin*x], [x^
2*sin*5*x, (A*x^2+B*x+C)*sin*5*x+(D*x^2+E*x+F)*cos*5*x], [x*exp(x)*cos
*2*x, (A*x+B)*exp(x)*cos*2*x+(C*x+D)*exp(x)*sin*2*x]]);" "6#-%'matrixG
6#7/7$-%\"fG6#%\"xG*(%S~~~~~~~~~~~~~~~~~Form~of~trial~particular~solut
ionG\"\"\"-%\"pG6#F+F.%,~~~~~~~~~~~GF.7$*&%$anyGF.%)constantGF.%\"AG7$
,&*&\"\")F.F+F.F.\"\"&!\"\",&*&F7F.F+F.F.%\"BGF.7$,(*$F+\"\"#F.*&\"\"(
F.F+F.F=F.F=,(*&F7F.*$F+FDF.F.*&F@F.F+F.F.%\"CGF.7$,(*&FDF.*$F+\"\"$F.
F.F+F=\"\"*F.,**&F7F.*$F+FPF.F.*&F@F.*$F+FDF.F.*&FKF.F+F.F.%\"DGF.7$*&
FPF.-%$expG6#*&\"\"%F.F+F.F.*&F7F.-Ffn6#*&FinF.F+F.F.7$**F<F.%$sinGF.F
PF.F+F.,&**F7F.%$cosGF.FPF.F+F.F.**F@F.F`oF.FPF.F+F.F.7$,$**FinF.FcoF.
FDF.F+F.F=,&**F7F.FcoF.FDF.F+F.F.**F@F.F`oF.FDF.F+F.F.7$*(FPF.F+F.-Ffn
6#,$F+F=F.*&,&*&F7F.F+F.F.F@F.F.-Ffn6#,$F+F=F.7$*&F+FD-Ffn6#*&F+F.FDF=
F.*&,(*&F7F.*$F+FDF.F.*&F@F.F+F.F.FKF.F.-Ffn6#*&F+F.FDF=F.7$*(-Ffn6#*&
FDF.F+F.F.F`oF.F+F.,&**F7F.-Ffn6#*&FDF.F+F.F.FcoF.F+F.F.**F@F.-Ffn6#*&
FDF.F+F.F.F`oF.F+F.F.7$**F+FDF`oF.F<F.F+F.,&**,(*&F7F.*$F+FDF.F.*&F@F.
F+F.F.FKF.F.F`oF.F<F.F+F.F.**,(*&FXF.*$F+FDF.F.*&%\"EGF.F+F.F.%\"FGF.F
.FcoF.F<F.F+F.F.7$*,F+F.-Ffn6#F+F.FcoF.FDF.F+F.,&*,,&*&F7F.F+F.F.F@F.F
.-Ffn6#F+F.FcoF.FDF.F+F.F.*,,&*&FKF.F+F.F.FXF.F.-Ffn6#F+F.F`oF.FDF.F+F
.F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "   ." }}{PARA 0 "" 
0 "" {TEXT 259 5 "Notes" }{TEXT -1 1 ":" }}{PARA 15 "" 0 "" {TEXT -1 
43 "This compilation is by no means exhaustive." }}{PARA 15 "" 0 "" 
{TEXT -1 46 "Sums of \"forcing\" functions of the given type " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 110 " should be given
 associated trial particular solutions which have the type of a sum of
 associated trial types." }}{PARA 15 "" 0 "" {TEXT -1 44 "It is assume
d in all of these examples that " }{TEXT 259 96 "no function of the tr
ial particular solution duplicates a function in the complementary sol
ution" }{TEXT -1 83 ". Modifications are necessary in such cases, as i
s shown by the following examples." }}{PARA 15 "" 0 "" {TEXT -1 5 "Whe
n " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 6 " is a " }
{TEXT 259 10 "polynomial" }{TEXT -1 15 " and c = 0 and " }{XPPEDIT 18 
0 "b<>0" "6#0%\"bG\"\"!" }{TEXT -1 139 ", the degree of the trial poly
nomial must be increased by 1.\nIf c = 0 and b = 0, the degree of the \+
trial polynomial must be increased by 2." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 9 "Example 4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 263 8 "Question" }{TEXT -1 2 ": " }
}{PARA 0 "" 0 "" {TEXT -1 56 "Find the general solution of the differe
ntial equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d
^2*y/(d*x^2)+4;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(
\"\"%F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+3*y = 4*exp(-3*x);" "6#
/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"$F'%\"yGF'F'*&\"\"%F'-%$expG6#,$*&F+
F'%\"xGF'F)F'" }{TEXT -1 16 "   ------- (i). " }{TEXT 261 0 "" }{TEXT 
-1 0 "" }}{PARA 257 "" 0 "" {TEXT 261 8 "Solution" }{TEXT 264 2 ": " }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The solution of the corre
sponding homogeneous equation: " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "d^2*y/(d*x^2)+4;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%
\"xGF&F(!\"\"F(\"\"%F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+3*y = 0;
" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"$F'%\"yGF'F'\"\"!" }{TEXT -1 
15 " ------- (ii), " }}{PARA 0 "" 0 "" {TEXT -1 50 "is obtained by con
sidering the auxiliary equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "m^2+4*m+3 = 0;" "6#/,(*$%\"mG\"\"#\"\"\"*&\"\"%F(F&F(F(
\"\"$F(\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 30 "This equ
ation is equivalent to" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "(m+1)*(m+3) = 0;" "6#/*&,&%\"mG\"\"\"F'F'F',&F&F'\"\"$F'F'\"\"!
" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 26 "and so has the solut
ions  " }{XPPEDIT 18 0 "m = -1;" "6#/%\"mG,$\"\"\"!\"\"" }{TEXT -1 5 "
 and " }{XPPEDIT 18 0 "m=-3" "6#/%\"mG,$\"\"$!\"\"" }{TEXT -1 2 ". " }
}{PARA 0 "" 0 "" {TEXT -1 43 "The general solution of (ii), which is t
he " }{TEXT 259 22 "complementary solution" }{TEXT -1 12 " of (i) is  \+
" }{XPPEDIT 18 0 "c(x) = C[1]*exp(-x)+C[2]*exp(-3*x);" "6#/-%\"cG6#%\"
xG,&*&&%\"CG6#\"\"\"F--%$expG6#,$F'!\"\"F-F-*&&F+6#\"\"#F--F/6#,$*&\"
\"$F-F'F-F2F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "deHG := diff(y(x),x$2)+4*dif
f(y(x),x)+3*y(x)=0;\ndsolve(deHG,y(x)):\nsubs(y(x)=c(x),%);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%deHGG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-
\"\"#\"\"\"*&\"\"%F2-F(6$F*F-F2F2*&\"\"$F2F*F2F2\"\"!" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%\"cG6#%\"xG,&*&%$_C1G\"\"\"-%$expG6#,$F'!\"\"F+
F+*&%$_C2GF+-F-6#,$*&\"\"$F+F'F+F0F+F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 111 "A problem arises if we attempt to find a particular solu
tion of (i) having the form of an exponential function " }{XPPEDIT 18 
0 "p(x) = A*exp(-3*x);" "6#/-%\"pG6#%\"xG*&%\"AG\"\"\"-%$expG6#,$*&\"
\"$F*F'F*!\"\"F*" }{TEXT -1 43 "as suggested by the form of the expres
sion " }{XPPEDIT 18 0 "4*exp(-3*x);" "6#*&\"\"%\"\"\"-%$expG6#,$*&\"\"
$F%%\"xGF%!\"\"F%" }{TEXT -1 27 " on the right side of (i). " }}{PARA 
0 "" 0 "" {TEXT -1 6 "Given " }{TEXT 302 1 "y" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "p(x) = A*exp(-3*x);" "6#/-%\"pG6#%\"xG*&%\"AG\"\"\"-%$e
xpG6#,$*&\"\"$F*F'F*!\"\"F*" }{TEXT -1 10 ", we have " }}{PARA 256 "" 
0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "dy/dx = p*`'`(x);" "6#/*&%#dyG\"
\"\"%#dxG!\"\"*&%\"pGF&-%\"'G6#%\"xGF&" }{XPPEDIT 18 0 "`` = -3*A*exp(
-3*x);" "6#/%!G,$*(\"\"$\"\"\"%\"AGF(-%$expG6#,$*&F'F(%\"xGF(!\"\"F(F0
" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = p*`''`(x);" "6#/*(
%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"*&%\"pGF(-%#''G6#F+F(" }
{XPPEDIT 18 0 "`` = 9*A*exp(-3*x);" "6#/%!G*(\"\"*\"\"\"%\"AGF'-%$expG
6#,$*&\"\"$F'%\"xGF'!\"\"F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 44 "Substituting the last three expressions for " }{XPPEDIT 18 0 "d
^2*y/(d*x^2)" "6#*(%\"dG\"\"#%\"yG\"\"\"*&F$F'*$%\"xGF%F'!\"\"" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }
{TEXT -1 5 " and " }{TEXT 303 1 "y" }{TEXT -1 28 " in (i) gives the eq
uation: " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "9*A*exp(
-3*x)+4*(-3*A*exp(-3*x))+3*A*exp(-3*x) = 4*exp(-3*x);" "6#/,(*(\"\"*\"
\"\"%\"AGF'-%$expG6#,$*&\"\"$F'%\"xGF'!\"\"F'F'*&\"\"%F',$*(F.F'F(F'-F
*6#,$*&F.F'F/F'F0F'F0F'F'*(F.F'F(F'-F*6#,$*&F.F'F/F'F0F'F'*&F2F'-F*6#,
$*&F.F'F/F'F0F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }
}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "0 = 4*exp(-3*x);" "
6#/\"\"!*&\"\"%\"\"\"-%$expG6#,$*&\"\"$F'%\"xGF'!\"\"F'" }{TEXT -1 2 "
. " }}{PARA 0 "" 0 "" {TEXT -1 116 "This equation is not valid for any
 real number x, and certainly does not give a possible value for the t
he constant " }{TEXT 304 1 "A" }{TEXT -1 9 " to make " }{XPPEDIT 18 0 
"y=A*exp(-3*x)" "6#/%\"yG*&%\"AG\"\"\"-%$expG6#,$*&\"\"$F'%\"xGF'!\"\"
F'" }{TEXT -1 20 " a solution of (i). " }}{PARA 0 "" 0 "" {TEXT -1 
128 "Thus it is impossible to find a partcular solution of (i) of this
 form. On reflection this should not be too surprising because " }
{XPPEDIT 18 0 "A*exp(-3*x)" "6#*&%\"AG\"\"\"-%$expG6#,$*&\"\"$F%%\"xGF
%!\"\"F%" }{TEXT -1 26 " has the form of the term " }{XPPEDIT 18 0 "C[
2]*exp(-3*x);" "6#*&&%\"CG6#\"\"#\"\"\"-%$expG6#,$*&\"\"$F(%\"xGF(!\"
\"F(" }{TEXT -1 76 " in the complementary solution. Hence we knew alre
ady that if we substitute " }{XPPEDIT 18 0 "y = C[2]*exp(-3*x);" "6#/%
\"yG*&&%\"CG6#\"\"#\"\"\"-%$expG6#,$*&\"\"$F*%\"xGF*!\"\"F*" }{TEXT 
-1 5 " or  " }{XPPEDIT 18 0 "y=A*exp(-3*x)" "6#/%\"yG*&%\"AG\"\"\"-%$e
xpG6#,$*&\"\"$F'%\"xGF'!\"\"F'" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "y=`
`" "6#/%\"yG%!G" }{TEXT -1 21 "\"any constant times\" " }{XPPEDIT 18 
0 "exp(-3*x)" "6#-%$expG6#,$*&\"\"$\"\"\"%\"xGF)!\"\"" }{TEXT -1 40 " \+
and the corresponding expressions for  " }{XPPEDIT 18 0 "d^2*y/(d*x^2)
" "6#*(%\"dG\"\"#%\"yG\"\"\"*&F$F'*$%\"xGF%F'!\"\"" }{TEXT -1 6 " and \+
 " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 63 "  \+
in the left side of (i), we get 0 rather than the right side " }
{XPPEDIT 18 0 "4*exp(-3*x)" "6#*&\"\"%\"\"\"-%$expG6#,$*&\"\"$F%%\"xGF
%!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 103 "The conclusion from this is that the initial gues
s as to the form of a possible particular solution is " }{TEXT 260 5 "
WRONG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 69 "The trouble her
e is that our initial guess for a particular solution " }{TEXT 259 51 
"duplicates a function in the complementary solution" }{TEXT -1 2 ". \+
" }}{PARA 0 "" 0 "" {TEXT -1 186 "In the light of the experience in lo
oking for the form of a general solution to a homogeneous equation in \+
the equal roots case we could try looking for a particular integral of
 the form " }{XPPEDIT 18 0 "p(x) = A*x*exp(-3*x);" "6#/-%\"pG6#%\"xG*(
%\"AG\"\"\"F'F*-%$expG6#,$*&\"\"$F*F'F*!\"\"F*" }{TEXT -1 19 ", that i
s, we have " }{TEXT 259 77 "multiplied the standard trial particular s
olution by the independent variable" }{TEXT -1 1 " " }{TEXT 300 1 "x" 
}{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{TEXT 301 1 "y
" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "p(x) = A*x*exp(-3*x);" "6#/-%\"pG6
#%\"xG*(%\"AG\"\"\"F'F*-%$expG6#,$*&\"\"$F*F'F*!\"\"F*" }{TEXT -1 9 ",
 we have" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "dy/dx = \+
p*`'`(x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%\"pGF&-%\"'G6#%\"xGF&" }
{TEXT -1 4 "  = " }{XPPEDIT 18 0 "A*exp(-3*x)-3*A*x*exp(-3*x) = A*exp(
-3*x)*(1-3*x);" "6#/,&*&%\"AG\"\"\"-%$expG6#,$*&\"\"$F'%\"xGF'!\"\"F'F
'**F-F'F&F'F.F'-F)6#,$*&F-F'F.F'F/F'F/*(F&F'-F)6#,$*&F-F'F.F'F/F',&F'F
'*&F-F'F.F'F/F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}
{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = p*`'
'`(x);" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"*&%\"pGF(-%#'
'G6#F+F(" }{XPPEDIT 18 0 "`` = -3*A*exp(-3*x)*(1-3*x)+A*exp(-3*x)*(-3)
;" "6#/%!G,&**\"\"$\"\"\"%\"AGF(-%$expG6#,$*&F'F(%\"xGF(!\"\"F(,&F(F(*
&F'F(F/F(F0F(F0*(F)F(-F+6#,$*&F'F(F/F(F0F(,$F'F0F(F(" }{TEXT -1 2 "  \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = A*exp(-3*x)*
(-3+9*x-3);" "6#/%!G*(%\"AG\"\"\"-%$expG6#,$*&\"\"$F'%\"xGF'!\"\"F',(F
-F/*&\"\"*F'F.F'F'F-F/F'" }{XPPEDIT 18 0 "`` = A*exp(-3*x)*(9*x-6);" "
6#/%!G*(%\"AG\"\"\"-%$expG6#,$*&\"\"$F'%\"xGF'!\"\"F',&*&\"\"*F'F.F'F'
\"\"'F/F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "Substitutin
g the last three expressions for " }{XPPEDIT 18 0 "d^2*y/(d*x^2)" "6#*
(%\"dG\"\"#%\"yG\"\"\"*&F$F'*$%\"xGF%F'!\"\"" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 5 " and " 
}{TEXT 305 1 "y" }{TEXT -1 28 " in (i) gives the equation: " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A*exp(-3*x)*(9*x-6)+4*A*exp
(-3*x)*(1-3*x)+3*A*x*exp(-3*x) = 4*exp(-3*x);" "6#/,(*(%\"AG\"\"\"-%$e
xpG6#,$*&\"\"$F'%\"xGF'!\"\"F',&*&\"\"*F'F.F'F'\"\"'F/F'F'**\"\"%F'F&F
'-F)6#,$*&F-F'F.F'F/F',&F'F'*&F-F'F.F'F/F'F'**F-F'F&F'F.F'-F)6#,$*&F-F
'F.F'F/F'F'*&F5F'-F)6#,$*&F-F'F.F'F/F'" }{TEXT -1 2 ", " }}{PARA 0 "" 
0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "A*exp(-3*x)*(9*x-6+4-12*x+3*x) = 4*exp(-3*x);" "6#/*(%
\"AG\"\"\"-%$expG6#,$*&\"\"$F&%\"xGF&!\"\"F&,,*&\"\"*F&F-F&F&\"\"'F.\"
\"%F&*&\"#7F&F-F&F.*&F,F&F-F&F&F&*&F3F&-F(6#,$*&F,F&F-F&F.F&" }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT 
-1 2 "  " }{XPPEDIT 18 0 "-2*A*exp(-3*x) = 4*exp(-3*x);" "6#/,$*(\"\"#
\"\"\"%\"AGF'-%$expG6#,$*&\"\"$F'%\"xGF'!\"\"F'F0*&\"\"%F'-F*6#,$*&F.F
'F/F'F0F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "Hence, taki
ng " }{XPPEDIT 18 0 "A = -2;" "6#/%\"AG,$\"\"#!\"\"" }{TEXT -1 32 ", g
ives the particular solution " }{XPPEDIT 18 0 "p(x) = -2*x*exp(-3*x);
" "6#/-%\"pG6#%\"xG,$*(\"\"#\"\"\"F'F+-%$expG6#,$*&\"\"$F+F'F+!\"\"F+F
2" }{TEXT -1 9 " for (i)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 46 "The general solution of (i) is the sum of the " }
{TEXT 259 22 "complementary solution" }}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{XPPEDIT 18 0 "c(x) = C[1]*exp(-x)+C[2]*exp(-3*x);" "6#/-%\"cG6#%
\"xG,&*&&%\"CG6#\"\"\"F--%$expG6#,$F'!\"\"F-F-*&&F+6#\"\"#F--F/6#,$*&
\"\"$F-F'F-F2F-F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "and t
he " }{TEXT 259 19 "particular solution" }{TEXT -1 1 " " }{XPPEDIT 18 
0 "p(x) = -2*x*exp(-3*x)" "6#/-%\"pG6#%\"xG,$*(\"\"#\"\"\"F'F+-%$expG6
#,$*&\"\"$F+F'F+!\"\"F+F2" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Hence the general solution of (
i) is:" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "y = C[1]*e
xp(-x)+C[2]*exp(-3*x)+(-2*x*exp(-3*x));" "6#/%\"yG,(*&&%\"CG6#\"\"\"F*
-%$expG6#,$%\"xG!\"\"F*F**&&F(6#\"\"#F*-F,6#,$*&\"\"$F*F/F*F0F*F*,$*(F
4F*F/F*-F,6#,$*&F9F*F/F*F0F*F0F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "de := diff(y(x),x
$2)+4*diff(y(x),x)+3*y(x)=4*exp(-3*x);\nunassign('A'):\npx := A*x*exp(
-3*x);\nsubs(y(x)=px,de);\neq := simplify(%);\nsolve(identity(eq,x),\{
A,B\});\np(x)=subs(%,px);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(
-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&\"\"%F2-F(6$F*F-F2F2*&\"
\"$F2F*F2F2,$*&F4F2-%$expG6#,$*&F8F2F-F2!\"\"F2F2" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#pxG*(%\"AG\"\"\"%\"xGF'-%$expG6#,$*&\"\"$F'F(F'!\"\"
F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%%diffG6$*(%\"AG\"\"\"%\"xGF
*-%$expG6#,$*&\"\"$F*F+F*!\"\"F*-%\"$G6$F+\"\"#F**&\"\"%F*-F&6$F(F+F*F
***F1F*F)F*F+F*F,F*F*,$*&F8F*F,F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%#eqG/,$*(\"\"#\"\"\"%\"AGF)-%$expG6#,$*&\"\"$F)%\"xGF)!\"\"F)F2,$*&
\"\"%F)F+F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"AG!\"#/%\"BGF(
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG,$*(\"\"#\"\"\"F'F+-
%$expG6#,$*&\"\"$F+F'F+!\"\"F+F2" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 " Dividing both sides of t
his equation by " }{XPPEDIT 18 0 "exp(x);" "6#-%$expG6#%\"xG" }{TEXT 
-1 37 ", (which is never zero), we see that " }{XPPEDIT 18 0 "A = -8/3
;" "6#/%\"AG,$*&\"\")\"\"\"\"\"$!\"\"F*" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "The general solution
 of (N) is the sum of the " }{TEXT 259 22 "complementary solution" }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c(x) = C[1]*exp(x)+C[
2]*exp(4*x);" "6#/-%\"cG6#%\"xG,&*&&%\"CG6#\"\"\"F--%$expG6#F'F-F-*&&F
+6#\"\"#F--F/6#*&\"\"%F-F'F-F-F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 8 "and the " }{TEXT 259 19 "particular solution" }{TEXT -1 2 
"  " }{XPPEDIT 18 0 "p(x) = -8*x*exp(x)/3;" "6#/-%\"pG6#%\"xG,$**\"\")
\"\"\"F'F+-%$expG6#F'F+\"\"$!\"\"F0" }{TEXT -1 1 "." }}{PARA 256 "" 0 
"" {TEXT -1 2 "\n " }{XPPEDIT 18 0 "y(x) = C[1]*exp(x)+C[2]*exp(4*x)-8
*x*exp(x)/3;" "6#/-%\"yG6#%\"xG,(*&&%\"CG6#\"\"\"F--%$expG6#F'F-F-*&&F
+6#\"\"#F--F/6#*&\"\"%F-F'F-F-F-**\"\")F-F'F--F/6#F'F-\"\"$!\"\"F>" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 69 "de := diff(y(x),x$2)-5*diff(y(x),x)+4*y(x)=8*exp
(x);\ndsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%
diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&\"\"&F2-F(6$F*F-F2!\"\"*&\"
\"%F2F*F2F2,$-%$expGF,\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG
6#%\"xG,(*&F'\"\"\"-%$expGF&F*#!\")\"\"$*&%$_C1GF*F+F*F**&%$_C2GF*-F,6
#,$F'\"\"%F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 1 ":" }}
{PARA 0 "" 0 "" {TEXT -1 67 "Our success here is an example of the fol
lowing general principles:" }}{PARA 15 "" 0 "" {TEXT 259 104 "No funct
ion in an assumed particular solution should duplicate a function in t
he complementary solution." }}{PARA 15 "" 0 "" {TEXT 259 207 "In order
 to avoid duplication of a function, multiply the standard trial parti
cular solution associated with a given forcing function by the smalles
t positve power of x which will eliminate that duplication." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 
}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 265 8 "Question" }
{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 60 "(a) Find the general so
lution of the differential equation: " }}{PARA 256 "" 0 "" {XPPEDIT 
18 0 "d^2*y/(d*x^2)+4*y = 12*cos*2*x;" "6#/,&*(%\"dG\"\"#%\"yG\"\"\"*&
F&F)*$%\"xGF'F)!\"\"F)*&\"\"%F)F(F)F)**\"#7F)%$cosGF)F'F)F,F)" }{TEXT 
-1 16 "   ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 74 "(b) Find the p
articular solution of (i) subject to the initial conditions " }
{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "y*`'`(0) = 6;" "6#/*&%\"yG\"\"\"-%\"'G6#\"\"!F&\"\"'" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" 
{TEXT 261 8 "Solution" }{TEXT 266 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 
60 "(a) The solution of the corresponding homogeneous equation: " }}
{PARA 256 "" 0 "" {XPPEDIT 18 0 "d^2*y/(d*x^2)+4*y = 0;" "6#/,&*(%\"dG
\"\"#%\"yG\"\"\"*&F&F)*$%\"xGF'F)!\"\"F)*&\"\"%F)F(F)F)\"\"!" }{TEXT 
-1 16 "   ------- (ii) " }}{PARA 0 "" 0 "" {TEXT -1 51 "is obtained by
 considering the auxiliary equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "m^2+4 = 0;" "6#/,&*$%\"mG\"\"#\"\"\"\"\"%F(\"\"!" 
}{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "This equation is equiv
alent to:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m^2 = -4
;" "6#/*$%\"mG\"\"#,$\"\"%!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" 
{TEXT -1 25 "and so has the solutions " }{XPPEDIT 18 0 "m =``" "6#/%\"
mG%!G" }{TEXT 306 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "2*i" "6#*&\"\"
#\"\"\"%\"iGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 43 "The ge
neral solution of (ii), which is the " }{TEXT 259 22 "complementary so
lution" }{TEXT -1 12 " of (i) is  " }{XPPEDIT 18 0 "c(x) = C[1]*cos*2*
x+C[2]*sin*2*x;" "6#/-%\"cG6#%\"xG,&**&%\"CG6#\"\"\"F-%$cosGF-\"\"#F-F
'F-F-**&F+6#F/F-%$sinGF-F/F-F'F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "deHG := di
ff(y(x),x$2)+4*y(x)=0;\ndsolve(deHG,y(x)):\nsubs(y(x)=c(x),%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%deHGG/,&-%%diffG6$-%\"yG6#%\"xG-%\"
$G6$F-\"\"#\"\"\"*&\"\"%F2F*F2F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/-%\"cG6#%\"xG,&*&%$_C1G\"\"\"-%$cosG6#,$F'\"\"#F+F+*&%$_C2GF+-%$si
nGF.F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "A trial particular
 solution of the form " }{XPPEDIT 18 0 "y=p(x)" "6#/%\"yG-%\"pG6#%\"xG
" }{XPPEDIT 18 0 "``=A*cos*2*x+B*sin*2*x" "6#/%!G,&**%\"AG\"\"\"%$cosG
F(\"\"#F(%\"xGF(F(**%\"BGF(%$sinGF(F*F(F+F(F(" }{TEXT -1 52 "  initial
ly suggested by the form of the expression " }{XPPEDIT 18 0 "12*cos*2*
x" "6#**\"#7\"\"\"%$cosGF%\"\"#F%%\"xGF%" }{TEXT -1 198 " on the right
 side of (i) exactly duplicates the complementary solution and so will
 not serve the desired purpose. It will only give 0 when substituted a
long with the corresponding expressions for  " }{XPPEDIT 18 0 "d^2*y/(
d*x^2)" "6#*(%\"dG\"\"#%\"yG\"\"\"*&F$F'*$%\"xGF%F'!\"\"" }{TEXT -1 6 
" and  " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 
27 "  in the left side of (i). " }}{PARA 0 "" 0 "" {TEXT -1 61 "Becaus
e of this an appropriate trial particular solution is  " }{XPPEDIT 18 
0 "y=p(x)" "6#/%\"yG-%\"pG6#%\"xG" }{XPPEDIT 18 0 "``=A*x*cos*2*x+B*x*
sin*2*x" "6#/%!G,&*,%\"AG\"\"\"%\"xGF(%$cosGF(\"\"#F(F)F(F(*,%\"BGF(F)
F(%$sinGF(F+F(F)F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "T
hen " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 "
   " }{XPPEDIT 18 0 "dy/dx = p*`'`(x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%
\"pGF&-%\"'G6#%\"xGF&" }{XPPEDIT 18 0 "`` = A*cos*2*x+A*x*(-2*sin*2*x)
+B*sin*2*x+B*x*``(2*cos*2*x);" "6#/%!G,***%\"AG\"\"\"%$cosGF(\"\"#F(%
\"xGF(F(*(F'F(F+F(,$**F*F(%$sinGF(F*F(F+F(!\"\"F(F(**%\"BGF(F/F(F*F(F+
F(F(*(F2F(F+F(-F$6#**F*F(F)F(F*F(F+F(F(F(" }{TEXT -1 1 " " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(A+2*B*x)*cos*2*x+(B-2*A
*x)*sin*2*x" "6#/%!G,&**,&%\"AG\"\"\"*(\"\"#F)%\"BGF)%\"xGF)F)F)%$cosG
F)F+F)F-F)F)**,&F,F)*(F+F)F(F)F-F)!\"\"F)%$sinGF)F+F)F-F)F)" }{TEXT 
-1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" 
{TEXT -1 3 "   " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = p*`''`(x);" "6#/*(%\"
dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"*&%\"pGF(-%#''G6#F+F(" }
{XPPEDIT 18 0 "``=2*B*cos*2*x+(A+2*B*x)*(-2*sin*2*x)+(-2*A)*sin*2*x+(B
-2*A*x)*``(2*cos*2*x)" "6#/%!G,**,\"\"#\"\"\"%\"BGF(%$cosGF(F'F(%\"xGF
(F(*&,&%\"AGF(*(F'F(F)F(F+F(F(F(,$**F'F(%$sinGF(F'F(F+F(!\"\"F(F(**,$*
&F'F(F.F(F3F(F2F(F'F(F+F(F(*&,&F)F(*(F'F(F.F(F+F(F3F(-F$6#**F'F(F*F(F'
F(F+F(F(F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "``= (4*B-4*A*x)*cos*2*x-(4*A+4*B*x)*sin*2*x" "6#/%!G,&*
*,&*&\"\"%\"\"\"%\"BGF*F**(F)F*%\"AGF*%\"xGF*!\"\"F*%$cosGF*\"\"#F*F.F
*F***,&*&F)F*F-F*F**(F)F*F+F*F.F*F*F*%$sinGF*F1F*F.F*F/" }{TEXT -1 2 "
. " }}{PARA 0 "" 0 "" {TEXT -1 17 "Substituting for " }{XPPEDIT 18 0 "
d^2*y/(d*x^2)" "6#*(%\"dG\"\"#%\"yG\"\"\"*&F$F'*$%\"xGF%F'!\"\"" }
{TEXT -1 5 " and " }{TEXT 307 1 "y" }{TEXT -1 28 " in (i) gives the eq
uation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(4*B-4*A*
x)*cos*2*x-(4*A+4*B*x)*sin*2*x+4*(A*x*cos*2*x+B*x*sin*2*x)=12*cos*2*x
" "6#/,(**,&*&\"\"%\"\"\"%\"BGF)F)*(F(F)%\"AGF)%\"xGF)!\"\"F)%$cosGF)
\"\"#F)F-F)F)**,&*&F(F)F,F)F)*(F(F)F*F)F-F)F)F)%$sinGF)F0F)F-F)F.*&F(F
),&*,F,F)F-F)F/F)F0F)F-F)F)*,F*F)F-F)F5F)F0F)F-F)F)F)F)**\"#7F)F/F)F0F
)F-F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "4*B*cos*2*x-4*A*sin*2
*x=12*cos*2*x" "6#/,&*,\"\"%\"\"\"%\"BGF'%$cosGF'\"\"#F'%\"xGF'F'*,F&F
'%\"AGF'%$sinGF'F*F'F+F'!\"\"**\"#7F'F)F'F*F'F+F'" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Equating \+
the coefficients of " }{XPPEDIT 18 0 "cos*2*x" "6#*(%$cosG\"\"\"\"\"#F
%%\"xGF%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sin*2*x" "6#*(%$sinG\"\"
\"\"\"#F%%\"xGF%" }{TEXT -1 52 " on the left and right sides of this e
quation gives " }{XPPEDIT 18 0 "A=0" "6#/%\"AG\"\"!" }{TEXT -1 5 " and
 " }{XPPEDIT 18 0 "B=3" "6#/%\"BG\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "This gives the parti
cular solution " }{XPPEDIT 18 0 "p(x) = 3*x*sin*2*x;" "6#/-%\"pG6#%\"x
G*,\"\"$\"\"\"F'F*%$sinGF*\"\"#F*F'F*" }{TEXT -1 11 "  for (i). " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "The gener
al solution of (i) is the sum of the " }{TEXT 259 22 "complementary so
lution" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "c(x) = C[1]*cos*2*x+C[2]*sin*
2*x;" "6#/-%\"cG6#%\"xG,&**&%\"CG6#\"\"\"F-%$cosGF-\"\"#F-F'F-F-**&F+6
#F/F-%$sinGF-F/F-F'F-F-" }{TEXT -1 9 " and the " }{TEXT 259 19 "partic
ular solution" }{TEXT -1 1 " " }{XPPEDIT 18 0 "p(x) = 3*x*sin*2*x" "6#
/-%\"pG6#%\"xG*,\"\"$\"\"\"F'F*%$sinGF*\"\"#F*F'F*" }{TEXT -1 2 ". " }
}{PARA 0 "" 0 "" {TEXT -1 42 "The general solution of (i) is therefore
: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=C[1]*cos*2*x+
C[2]*sin*2*x+3*x*sin*2*x" "6#/%\"yG,(**&%\"CG6#\"\"\"F*%$cosGF*\"\"#F*
%\"xGF*F***&F(6#F,F*%$sinGF*F,F*F-F*F**,\"\"$F*F-F*F1F*F,F*F-F*F*" }
{TEXT -1 16 " ------- (iii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "de := diff(y(x),x$2)+4*y(x)
=12*cos(2*x);\nunassign('A','B'):\npx := A*x*cos(2*x)+B*x*sin(2*x);\ns
ubs(y(x)=px,de);\neq := simplify(%);\nsolve(identity(eq,x),\{A,B\});\n
p(x)=subs(%,px);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6
$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&\"\"%F2F*F2F2,$*&\"#7F2-%$cosG6#,
$*&F1F2F-F2F2F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#pxG,&*(%\"AG\"
\"\"%\"xGF(-%$cosG6#,$*&\"\"#F(F)F(F(F(F(*(%\"BGF(F)F(-%$sinGF,F(F(" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%%diffG6$,&*(%\"AG\"\"\"%\"xGF+-%
$cosG6#,$*&\"\"#F+F,F+F+F+F+*(%\"BGF+F,F+-%$sinGF/F+F+-%\"$G6$F,F2F+**
\"\"%F+F*F+F,F+F-F+F+**F;F+F4F+F,F+F5F+F+,$*&\"#7F+F-F+F+" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#eqG/,&*(\"\"%\"\"\"%\"AGF)-%$sinG6#,$*&\"\"#
F)%\"xGF)F)F)!\"\"*(F(F)%\"BGF)-%$cosGF-F)F),$*&\"#7F)F5F)F)" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#<$/%\"BG\"\"$/%\"AG\"\"!" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%\"pG6#%\"xG,$*(\"\"$\"\"\"F'F+-%$sinG6#,$*&\"\"#F+
F'F+F+F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 39 "We can find the general solution using " }{TEXT 0 6 "dsolve" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 57 "de := diff(y(x),x$2)+4*y(x)=12*cos(2*x);\ndsolve
(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"y
G6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&\"\"%F2F*F2F2,$*&\"#7F2-%$cosG6#,$*&F1F
2F-F2F2F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&-%$si
nG6#,$*&\"\"#\"\"\"F'F0F0F0%$_C2GF0F0*&-%$cosGF,F0%$_C1GF0F0*(\"\"$F0F
'F0F*F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "(b) Suppose now th
at we have the initial conditions " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%
\"yG6#\"\"!F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y*`'`(0) = 6;" "6#/
*&%\"yG\"\"\"-%\"'G6#\"\"!F&\"\"'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 6 "Since " }{XPPEDIT 18 0 "y = 0;" "6#/%\"yG\"\"!" }{TEXT -1 
6 " when " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 43 ", substi
tuting these values in (iii) gives " }{XPPEDIT 18 0 "0 = C[1];" "6#/\"
\"!&%\"CG6#\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "Sub
stituting " }{XPPEDIT 18 0 "C[1]=0" "6#/&%\"CG6#\"\"\"\"\"!" }{TEXT 
-1 26 ", equation (iii) becomes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " 
}{XPPEDIT 18 0 "y=(C[2]+3*x)*sin*2*x" "6#/%\"yG**,&&%\"CG6#\"\"#\"\"\"
*&\"\"$F+%\"xGF+F+F+%$sinGF+F*F+F.F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 
"" {TEXT -1 9 "so that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "dy/dx=y*`'`(x)" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%\"yGF&-%
\"'G6#%\"xGF&" }{XPPEDIT 18 0 "`` = 3*sin*2*x+(C[2]+3*x)*2*cos*2*x;" "
6#/%!G,&**\"\"$\"\"\"%$sinGF(\"\"#F(%\"xGF(F(*,,&&%\"CG6#F*F(*&F'F(F+F
(F(F(F*F(%$cosGF(F*F(F+F(F(" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" 
{TEXT -1 6 "Since " }{XPPEDIT 18 0 "dy/dx = 6;" "6#/*&%#dyG\"\"\"%#dxG
!\"\"\"\"'" }{TEXT -1 7 ", when " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!
" }{TEXT -1 26 ", we obtain the equation: " }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "6 = 2*C[2];" "6#/\"\"'*&\"\"#\"\"\"&%\"CG6#F&
F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 
18 0 "C[2]=3" "6#/&%\"CG6#\"\"#\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "The particular solutio
n satisfying the initial conditions " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-
%\"yG6#\"\"!F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y*`'`(0) = 6;" "6#
/*&%\"yG\"\"\"-%\"'G6#\"\"!F&\"\"'" }{TEXT -1 6 "  is: " }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = (3*x+3)*sin*2*x;" "6#/%\"yG*
*,&*&\"\"$\"\"\"%\"xGF)F)F(F)F)%$sinGF)\"\"#F)F*F)" }{TEXT -1 2 ". " }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 245 "de := diff(y(x),x$2)+4*y(x)=12*cos(2*x);\nic := y(0)=0,D(y)(0)=
6;\ndsolve(\{de,ic\},y(x));\ng := unapply(rhs(%),x);\np1 := plot(g(x),
x=0..14.2):\np2 := plot([3*x+3,-3*x-3],x=0..14.2,color=black,linestyle
=2):\nplots[display]([p1,p2],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"
*&\"\"%F2F*F2F2,$*&\"#7F2-%$cosG6#,$*&F1F2F-F2F2F2F2" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)\"\"'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&\"\"$\"\"\"-%$sinG6#
,$*&\"\"#F+F'F+F+F+F+*(F*F+F'F+F,F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&\"\"$\"\"\"-%$sinG6#,$
*&\"\"#F/9$F/F/F/F/*(F.F/F6F/F0F/F/F(F(F(" }}{PARA 13 "" 1 "" 
{GLPLOT2D 479 349 349 {PLOTDATA 2 "6'-%'CURVESG6$7jt7$$\"\"!F)F(7$$\"3
kmmm\"*e>&4$!#=$\"39V_C>wbzA!#<7$$\"3S+]P*e_<W%F-$\"3=Q'4Nv!=iLF07$$\"
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%y(x)G-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F($\"$U\"F^]nFi`o" 1 2 0 1 10 0 
2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+
3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "The magnitude of the osci
llations of the graph increase steadily. " }}{PARA 0 "" 0 "" {TEXT -1 
48 "The curve lies between the straight line graphs " }{XPPEDIT 18 0 "
y=3*x+3" "6#/%\"yG,&*&\"\"$\"\"\"%\"xGF(F(F'F(" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "y=-3*x-3" "6#/%\"yG,&*&\"\"$\"\"\"%\"xGF(!\"\"F'F*" }
{TEXT -1 12 ".  The line " }{XPPEDIT 18 0 "y=3*x+3" "6#/%\"yG,&*&\"\"$
\"\"\"%\"xGF(F(F'F(" }{TEXT -1 44 " is tangential to the curve at poin
ts where " }{XPPEDIT 18 0 "x = Pi/4+k*Pi;" "6#/%\"xG,&*&%#PiG\"\"\"\"
\"%!\"\"F(*&%\"kGF(F'F(F(" }{TEXT -1 3 "  (" }{TEXT 308 1 "k" }{TEXT 
-1 85 " an integer), just to the left the various maximum points on th
e curve, and the line " }{XPPEDIT 18 0 "y=-3*x-3" "6#/%\"yG,&*&\"\"$\"
\"\"%\"xGF(!\"\"F'F*" }{TEXT -1 44 " is tangential to the curve at poi
nts where " }{XPPEDIT 18 0 "x = 3*Pi/4+k*Pi;" "6#/%\"xG,&*(\"\"$\"\"\"
%#PiGF(\"\"%!\"\"F(*&%\"kGF(F)F(F(" }{TEXT -1 2 " (" }{TEXT 309 1 "k" 
}{TEXT -1 51 " an integer), just to the left the minimum points. " }}
{PARA 0 "" 0 "" {TEXT -1 102 "In applications to physical systems invo
lving oscillations solutions like this model the phenomena of " }
{TEXT 259 9 "resonance" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 9 "Example 6" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}{PARA 0 "" 0 "" {TEXT 267 8 "Question" }{TEXT -1 2 ": " }}
{PARA 0 "" 0 "" {TEXT -1 54 "Find the general solution of the differen
tial equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{XPPEDIT 18 0 "d^2*y/(d*x^2)-2;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%
\"xGF&F(!\"\"F(F&F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+y = exp(x);
" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'%\"yGF'-%$expG6#%\"xG" }{TEXT -1 13 "
  ------- (i)" }}{PARA 257 "" 0 "" {TEXT 261 8 "Solution" }{TEXT 268 
2 ": " }}{PARA 0 "" 0 "" {TEXT -1 54 "The solution of the correspondin
g homogeneous equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "
" 0 "" {XPPEDIT 18 0 "d^2*y/(d*x^2)-2;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&
F%F(*$%\"xGF&F(!\"\"F(F&F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+y = \+
0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'%\"yGF'\"\"!" }{TEXT -1 15 "   ----
--- (ii)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
49 "is obtained by considering the auxiliary equation" }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m^2-2*m+1 = 0;" "6#/,(*$%\"mG\"\"#
\"\"\"*&F'F(F&F(!\"\"F(F(\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "(m
-1)^2 = 0;" "6#/*$,&%\"mG\"\"\"F'!\"\"\"\"#\"\"!" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 7 "giving " }{XPPEDIT 18 0 "m = 1;" "6#/%\"mG
\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 42 "The general solution of (ii) which is the " }{TEXT 
259 22 "complementary solution" }{TEXT -1 10 " of (i) is" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 
0 "c(x) = C[1]*exp(x)+C[2]*x*exp(x);" "6#/-%\"cG6#%\"xG,&*&&%\"CG6#\"
\"\"F--%$expG6#F'F-F-*(&F+6#\"\"#F-F'F--F/6#F'F-F-" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
83 "deHG := diff(y(x),x$2)-2*diff(y(x),x)+y(x)=0;\ndsolve(deHG,y(x)):
\nsubs(y(x)=c(x),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%deHGG/,(-%%
diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2!\"\"F*F2\"
\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"cG6#%\"xG,&*&%$_C1G\"\"\"-
%$expGF&F+F+*(%$_C2GF+F,F+F'F+F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 65 "It will be no use looking for a particular solution of th
e form  " }{XPPEDIT 18 0 "p(x) = A*exp(x);" "6#/-%\"pG6#%\"xG*&%\"AG\"
\"\"-%$expG6#F'F*" }{TEXT -1 10 ", or even " }{XPPEDIT 18 0 "p(x) = A*
x*exp(x);" "6#/-%\"pG6#%\"xG*(%\"AG\"\"\"F'F*-%$expG6#F'F*" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 "because the functions " }
{XPPEDIT 18 0 "exp(x);" "6#-%$expG6#%\"xG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "x*exp(x);" "6#*&%\"xG\"\"\"-%$expG6#F$F%" }{TEXT -1 46 
" already appear in the complementary solution." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Instead let's try to find
 a particular solution of the form " }{XPPEDIT 18 0 "p(x) = A*x^2*exp(
x);" "6#/-%\"pG6#%\"xG*(%\"AG\"\"\"*$F'\"\"#F*-%$expG6#F'F*" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 110 "de := diff(y(x),x$2)-2*diff(y(x),x)+y(x)=exp(x);\nun
assign('A'):\nsubs(y(x)=A*x^2*exp(x),de);\neq := simplify(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G
6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2!\"\"F*F2-%$expGF," }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/,(-%%diffG6$*(%\"AG\"\"\")%\"xG\"\"#F*-%$expG6#F,F*-
%\"$G6$F,F-F**&F-F*-F&6$F(F,F*!\"\"F(F*F." }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#eqG/,$*&%\"AG\"\"\"-%$expG6#%\"xGF)\"\"#F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "We can obtain a particular solu
tion by taking  " }{XPPEDIT 18 0 "A = 1/2;" "6#/%\"AG*&\"\"\"F&\"\"#!
\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 46 "The general solution of (N) is the sum of the " }{TEXT 
259 22 "complementary solution" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "c(x) \+
= C[1]*exp(x)+C[2]*x*exp(x);" "6#/-%\"cG6#%\"xG,&*&&%\"CG6#\"\"\"F--%$
expG6#F'F-F-*(&F+6#\"\"#F-F'F--F/6#F'F-F-" }{TEXT -1 1 " " }}{PARA 0 "
" 0 "" {TEXT -1 8 "and the " }{TEXT 259 19 "particular solution" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "p(x) = x^2*exp(x)/2;" "6#/-%\"pG6#%\"xG
*(F'\"\"#-%$expG6#F'\"\"\"F)!\"\"" }{TEXT -1 1 "." }}{PARA 256 "" 0 "
" {TEXT -1 2 "\n " }{XPPEDIT 18 0 "y(x) = C[1]*exp(x)+C[2]*x*exp(x)+x^
2*exp(x)/2;" "6#/-%\"yG6#%\"xG,(*&&%\"CG6#\"\"\"F--%$expG6#F'F-F-*(&F+
6#\"\"#F-F'F--F/6#F'F-F-*(F'F4-F/6#F'F-F4!\"\"F-" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
65 "de := diff(y(x),x$2)-2*diff(y(x),x)+y(x)=exp(x);\ndsolve(de,y(x));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%
\"$G6$F-\"\"#\"\"\"-F(6$F*F-!\"#F*F2-%$expGF," }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&)F'\"\"#\"\"\"-%$expGF&F,#F,F+*&%$_C1
GF,F-F,F,*(%$_C2GF,F-F,F'F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "I
nstructions and example" }}{PARA 0 "" 0 "" {TEXT -1 110 "In questions \+
1 to 11, find the general solution of the given differential equations
 using the following steps:" }}{PARA 0 "" 0 "" {TEXT -1 8 "(a) Use " }
{TEXT 0 6 "dsolve" }{TEXT -1 36 " to find the complementary solution.
" }}{PARA 0 "" 0 "" {TEXT -1 78 "(b) Use the method of undetermined co
efficients to find a particular solution." }}{PARA 0 "" 0 "" {TEXT -1 
70 "(c) Check your combined general solution by substitution, or by us
ing " }{TEXT 0 6 "dsolve" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 257 "" 0 "" {TEXT 261 7 "Example" }{TEXT 269 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "   2 " }
{XPPEDIT 18 0 "d^2*y/(d*x^2)+3;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%
\"xGF&F(!\"\"F(\"\"$F(" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "dy/dx-2*y = 1
4*x^2-4*x-11;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"#F'%\"yGF'F),(*&\"
#9F'*$%\"xGF+F'F'*&\"\"%F'F1F'F)\"#6F)" }{TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 270 8 "Solution" }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 
"(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "deHG := 2*diff(y(x),
x$2)+3*diff(y(x),x)-2*y(x)=0;\ndsolve(deHG,y(x));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%deHGG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#F1*&\"
\"$\"\"\"-F(6$F*F-F4F4*&F1F4F*F4!\"\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&%$_C1G\"\"\"-%$expG6#,$F'!\"#F+F+*&%$
_C2GF+-F-6#,$F'#F+\"\"#F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 68 "(b) Set up the appropriate trial particular sol
ution, in this case  " }{XPPEDIT 18 0 "p(x)=A*x^2+B*x+C" "6#/-%\"pG6#%
\"xG,(*&%\"AG\"\"\"*$F'\"\"#F+F+*&%\"BGF+F'F+F+%\"CGF+" }{TEXT -1 43 "
, and find the \"undetermined coefficients\" " }{XPPEDIT 18 0 "A, B" "
6$%\"AG%\"BG" }{TEXT -1 5 " and " }{TEXT 271 1 "C" }{TEXT -1 18 ", by \+
substituting " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 55 " \+
in the original non-homogeneous differential equation." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "de := \+
2*diff(y(x),x$2)+3*diff(y(x),x)-2*y(x)=14*x^2-4*x-11;\nunassign('A','B
','C'):\nsubs(y(x)=A*x^2+B*x+C,de);\neq := simplify(%);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#F
1*&\"\"$\"\"\"-F(6$F*F-F4F4*&F1F4F*F4!\"\",(*$)F-F1F4\"#9*&\"\"%F4F-F4
F8\"#6F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,-%%diffG6$,(*&%\"AG\"\"
\")%\"xG\"\"#F+F+*&%\"BGF+F-F+F+%\"CGF+-%\"$G6$F-F.F.*&\"\"$F+-F&6$F(F
-F+F+*(F.F+F*F+F,F+!\"\"*(F.F+F0F+F-F+F:*&F.F+F1F+F:,(*$F,F+\"#9*&\"\"
%F+F-F+F:\"#6F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/,.%\"AG\"\"%
*(\"\"'\"\"\"F'F+%\"xGF+F+*&\"\"$F+%\"BGF+F+*(\"\"#F+F'F+)F,F1F+!\"\"*
(F1F+F/F+F,F+F3*&F1F+%\"CGF+F3,(*$F2F+\"#9*&F(F+F,F+F3\"#6F3" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
56 "solve(identity(eq,x),\{A,B,C\});\np(x)=subs(%,A*x^2+B*x+C);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"AG!\"(/%\"BG!#>/%\"CG!#P" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG,(*$)F'\"\"#\"\"\"!\"(*&
\"#>F,F'F,!\"\"\"#PF0" }}}{PARA 0 "" 0 "" {TEXT -1 27 " The general so
lution is:  " }{XPPEDIT 18 0 "y(x) = C[1]*exp(-2*x)+C[2]*exp(x/2)-7*x^
2-19*x-37;" "6#/-%\"yG6#%\"xG,,*&&%\"CG6#\"\"\"F--%$expG6#,$*&\"\"#F-F
'F-!\"\"F-F-*&&F+6#F3F--F/6#*&F'F-F3F4F-F-*&\"\"(F-*$F'F3F-F4*&\"#>F-F
'F-F4\"#PF4" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 26 "Check by substitution . . " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "subs(y(x)=C[
1]*exp(-2*x)+C[2]*exp(x/2)-7*x^2-19*x-37,de);\nsimplify(%);" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#/,0-%%diffG6$,,*&&%\"CG6#\"\"\"F--%$expG6#,$
%\"xG!\"#F-F-*&&F+6#\"\"#F--F/6#,$F2#F-F7F-F-*&\"\"(F-)F2F7F-!\"\"*&\"
#>F-F2F-F?\"#PF?-%\"$G6$F2F7F7*&\"\"$F--F&6$F(F2F-F-*(F7F-F*F-F.F-F?*(
F7F-F5F-F8F-F?*&\"#9F-F>F-F-*&\"#QF-F2F-F-\"#uF-,(*$F>F-FM*&\"\"%F-F2F
-F?\"#6F?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$)%\"xG\"\"#\"\"\"\"#
9*&\"\"%F)F'F)!\"\"\"#6F-F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 22 " . . . or check using " }{TEXT 0 6 "dsolve" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 16 "dsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%\"yG6#%\"xG,,*$)F'\"\"#\"\"\"!\"(F'!#>!#PF,*&%$_C1GF,-%$expG6
#,$F'!\"#F,F,*&%$_C2GF,-F36#,$F'#F,F+F,F," }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 
"" {TEXT -1 52 "____________________________________________________" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 3 "Q1 " }}{PARA 258 "" 0 "
" {TEXT -1 6 "      " }{XPPEDIT 18 0 "d^2*y/(d*x^2)-2;" "6#,&*(%\"dG\"
\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(F&F," }{TEXT -1 1 " " }{XPPEDIT 
18 0 "dy/dx-8*y = 2*x^2-7*x-2" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\")F
'%\"yGF'F),(*&\"\"#F'*$%\"xGF/F'F'*&\"\"(F'F1F'F)F/F)" }{TEXT -1 1 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 45 "_____________________________________
_______ " }}{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 45 "____________________________________________ " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 3 "Q2 " }}{PARA 258 "" 0 "" {TEXT -1 6 "      " }{XPPEDIT 
18 0 "d^2*y/(d*x^2)+dy/dx-6*y = 2*x;" "6#/,(*(%\"dG\"\"#%\"yG\"\"\"*&F
&F)*$%\"xGF'F)!\"\"F)*&%#dyGF)%#dxGF-F)*&\"\"'F)F(F)F-*&F'F)F,F)" }
{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 45 "_______________________
_____________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 45 "_______________________________________
_____ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 258 "" 0 "" {TEXT -1 4 "    \+
" }{XPPEDIT 18 0 "d^2*y/(d*x^2)+2;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(
*$%\"xGF&F(!\"\"F(F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+2*y = 3*
exp(-2*x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"#F'%\"yGF'F'*&\"\"$F'
-%$expG6#,$*&F+F'%\"xGF'F)F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 45 "____________________________________________ " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 45 "____________
________________________________ " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }}
{PARA 258 "" 0 "" {TEXT -1 4 "    " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+dy/d
x-2*y = 5*sin(2*x);" "6#/,(*(%\"dG\"\"#%\"yG\"\"\"*&F&F)*$%\"xGF'F)!\"
\"F)*&%#dyGF)%#dxGF-F)*&F'F)F(F)F-*&\"\"&F)-%$sinG6#*&F'F)F,F)F)" }
{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 45 "_______________________
_____________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 45 "_______________________________________
_____ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }}{PARA 258 "" 0 "" {TEXT -1 4 "    \+
" }{XPPEDIT 18 0 "d^2*y/(d*x^2)+2*y = cos(3*x);" "6#/,&*(%\"dG\"\"#%\"
yG\"\"\"*&F&F)*$%\"xGF'F)!\"\"F)*&F'F)F(F)F)-%$cosG6#*&\"\"$F)F,F)" }
{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 45 "_______________________
_____________________ " }}{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 45 "_______________________________________
_____ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 3 "Q6 " }}{PARA 258 "" 0 "" {TEXT -1 5 "     \+
" }{XPPEDIT 18 0 "d^2*y/(d*x^2)+4;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(
*$%\"xGF&F(!\"\"F(\"\"%F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+4*y =
 8*x+exp(x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"%F'%\"yGF'F',&*&\"
\")F'%\"xGF'F'-%$expG6#F0F'" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT 
-1 45 "____________________________________________ " }}{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 45 "____________
________________________________ " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q7 " }}
{PARA 258 "" 0 "" {TEXT -1 4 "    " }{XPPEDIT 18 0 "d^2*y/(d*x^2)-4;" 
"6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(\"\"%F," }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "dy/dx+3*y = exp(3*x);" "6#/,&*&%#dyG\"\"\"%#d
xG!\"\"F'*&\"\"$F'%\"yGF'F'-%$expG6#*&F+F'%\"xGF'" }{TEXT -1 2 "  " }}
{PARA 0 "" 0 "" {TEXT -1 45 "_________________________________________
___ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 45 "____________________________________________ " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 3 "Q8 " }}{PARA 258 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 
18 0 "d^2*y/(d*x^2)+4*y = 4*sin(2*x);" "6#/,&*(%\"dG\"\"#%\"yG\"\"\"*&
F&F)*$%\"xGF'F)!\"\"F)*&\"\"%F)F(F)F)*&F/F)-%$sinG6#*&F'F)F,F)F)" }
{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 45 "_______________________
_____________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 45 "_______________________________________
_____ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 3 "Q9 " }}{PARA 258 "" 0 "" {TEXT -1 31 "Find
 the general solution of   " }{XPPEDIT 18 0 "4;" "6#\"\"%" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+4;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&
F%F(*$%\"xGF&F(!\"\"F(\"\"%F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx-3
*y = sin(x)-2*cos(x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"$F'%\"yGF'
F),&-%$sinG6#%\"xGF'*&\"\"#F'-%$cosG6#F1F'F)" }{TEXT -1 2 ". " }}
{PARA 258 "" 0 "" {TEXT -1 68 "Also find the particular solution subje
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