{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 262 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE " " 258 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2 " -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 } 1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Nor mal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 62 "Non-homogeneous linear 2nd order \+ DE's: variation of parameters" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 27.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 68 "Particular solution by t he method of variation of parameters: theory" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 78 "Consider the 2nd order linear diferential equation with 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(*$% \"xGF&F(!\"\"F(%\"bGF(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+c*y = f( x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&%\"cGF'%\"yGF'F'-%\"fG6#%\"xG" } {TEXT -1 13 " ------- (N)" }}{PARA 0 "" 0 "" {TEXT -1 77 "Suppose tha t we know the general solution of the related homogeneous equation" }} {PARA 256 "" 0 "" {XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 1 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2)+b;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$% \"xGF&F(!\"\"F(%\"bGF(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+c*y = 0; " "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&%\"cGF'%\"yGF'F'\"\"!" }{TEXT -1 13 " ------- (H)" }}{PARA 0 "" 0 "" {TEXT -1 2 "is" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y(x) = C[1]*h[1](x)+C[2]*h[2](x);" "6#/-%\"yG6#%\"xG,&*&&%\"CG6#\"\"\"F--&%\"hG6#F-6#F'F-F-*&&F+6#\"\"#F- -&F06#F66#F'F-F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 13 "whic h is the " }{TEXT 259 22 "complementary solution" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c(x) = C[1]*h[1](x)+ C[2]*h[2](x);" "6#/-%\"cG6#%\"xG,&*&&%\"CG6#\"\"\"F--&%\"hG6#F-6#F'F-F -*&&F+6#\"\"#F--&F06#F66#F'F-F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 " for (N)." }}{PARA 0 "" 0 "" {TEXT -1 187 "The three cases which can occur, depending on the nature of the roots of the auxiliar y equation associated with (H), all fit into this pattern, with approp riate choices for the functions " }{XPPEDIT 18 0 "h[1](x)" "6#-&%\"hG6 #\"\"\"6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "h[2](x)" "6#-&%\"h G6#\"\"#6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 22 "We look for a related " }{TEXT 259 19 "pa rticular solution" }{TEXT -1 20 " of (N) of the form " }{XPPEDIT 18 0 "x = p(x)" "6#/%\"xG-%\"pG6#F$" }{TEXT -1 6 " where" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "p( x) = u[1](x)*h[1](x)+u[2](x)*h[2](x);" "6#/-%\"pG6#%\"xG,&*&-&%\"uG6# \"\"\"6#F'F.-&%\"hG6#F.6#F'F.F.*&-&F,6#\"\"#6#F'F.-&F26#F96#F'F.F." } {TEXT -1 13 " ------- (P)," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 262 18 "__________________" }{TEXT -1 17 " " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 71 "with the pr oblem being that of determining the \"function coefficients\" " } {XPPEDIT 18 0 "u[1](x)" "6#-&%\"uG6#\"\"\"6#%\"xG" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "u[2](x)" "6#-&%\"uG6#\"\"#6#%\"xG" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "In orde r for us to be able to substitute the expression (P) in (N), we must c alculate its first and second derivatives." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx;" "6#* &%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[p(x)] = u[1]( x)*k[1](x)+u[2](x)*k[2](x)+[v[1](x)*h[1](x)+v[2](x)*h[2](x)];" "6#/7#- %\"pG6#%\"xG,(*&-&%\"uG6#\"\"\"6#F(F/-&%\"kG6#F/6#F(F/F/*&-&F-6#\"\"#6 #F(F/-&F36#F:6#F(F/F/7#,&*&-&%\"vG6#F/6#F(F/-&%\"hG6#F/6#F(F/F/*&-&FE6 #F:6#F(F/-&FJ6#F:6#F(F/F/F/" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "d/dx;" "6#*&%\"dG\"\"\"%#dxG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "[h[i](x)] = k[i](x);" "6#/7#-&%\"hG6#% \"iG6#%\"xG-&%\"kG6#F)6#F+" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "d/dx; " "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[u[i](x) ] = v[i](x);" "6#/7#-&%\"uG6#%\"iG6#%\"xG-&%\"vG6#F)6#F+" }{TEXT -1 5 " for " }{XPPEDIT 18 0 " i = 1,2" "6$/%\"iG\"\"\"\"\"#" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "We ar e not really interested in " }{XPPEDIT 18 0 "u[1](x);" "6#-&%\"uG6#\" \"\"6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u[2](x);" "6#-&%\"uG6 #\"\"#6#%\"xG" }{TEXT -1 52 " for their own sake, but only as a means \+ of finding " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 41 "It may well be that different choices of " }{XPPEDIT 18 0 "u[1](x);" "6#-&%\"uG6#\"\"\"6#%\"xG" }{TEXT -1 5 " \+ and " }{XPPEDIT 18 0 "u[2](x);" "6#-&%\"uG6#\"\"#6#%\"xG" }{TEXT -1 23 " will lead to the same " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" } {TEXT -1 38 ", so there is some freedom in picking " }{XPPEDIT 18 0 "u [1](x);" "6#-&%\"uG6#\"\"\"6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u[2](x);" "6#-&%\"uG6#\"\"#6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 41 "Let us assume that we can find functions " }{XPPEDIT 18 0 "u[1](x);" "6#-&%\"uG6#\"\"\"6#%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "u[2](x);" "6#-&%\"uG6#\"\"#6#%\"xG" }{TEXT -1 49 " so t hat the term in the square brackets is zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v[1](x)*h[1 ](x)+v[2](x)*h[2](x) = 0;" "6#/,&*&-&%\"vG6#\"\"\"6#%\"xGF*-&%\"hG6#F* 6#F,F*F**&-&F(6#\"\"#6#F,F*-&F/6#F66#F,F*F*\"\"!" }{TEXT -1 13 " ---- --- (A)" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "d/dx;" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[u[i](x)] \+ = v[i](x);" "6#/7#-&%\"uG6#%\"iG6#%\"xG-&%\"vG6#F)6#F+" }{TEXT -1 6 " \+ for " }{XPPEDIT 18 0 "i=1,2" "6$/%\"iG\"\"\"\"\"#" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx;" "6#*&%\"dG\"\" \"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[p(x)] = u[1](x)*k[1](x) +u[2](x)*k[2](x);" "6#/7#-%\"pG6#%\"xG,&*&-&%\"uG6#\"\"\"6#F(F/-&%\"kG 6#F/6#F(F/F/*&-&F-6#\"\"#6#F(F/-&F36#F:6#F(F/F/" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "In additi on to the equation (A) we obtain a second condition which the function s " }{XPPEDIT 18 0 "u[1](x);" "6#-&%\"uG6#\"\"\"6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u[2](x);" "6#-&%\"uG6#\"\"#6#%\"xG" }{TEXT -1 30 " must satisfy by substituting " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#% \"xG" }{TEXT -1 23 " into the equation (N)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "The following Maple code determ ines expressions for the 1st and 2nd derivatives of " }{XPPEDIT 18 0 " p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 35 ", making use of the assumption ( A)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "unassign('x','u','h','f','k','a','b','c'):\np := x - > u[1](x)*h[1](x)+u[2](x)*h[2](x):\np(x);\ndp := x -> u[1](x)*diff(h[1 ](x),x)+u[2](x)*diff(h[2](x),x):\ndp(x);\ndiff(dp(x),x);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&*&-&%\"uG6#\"\"\"6#%\"xGF)-&%\"hGF(F*F)F)*&-&F '6#\"\"#F*F)-&F.F2F*F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-&%\"u G6#\"\"\"6#%\"xGF)-%%diffG6$-&%\"hGF(F*F+F)F)*&-&F'6#\"\"#F*F)-F-6$-&F 1F5F*F+F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&-%%diffG6$-&%\"uG6# \"\"\"6#%\"xGF.F,-F&6$-&%\"hGF+F-F.F,F,*&F(F,-F&6$F1-%\"$G6$F.\"\"#F,F ,*&-F&6$-&F*6#F:F-F.F,-F&6$-&F3F@F-F.F,F,*&F>F,-F&6$FCF7F,F," }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Substitut e these derivatives in the differential equation (N).\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "eq1 := a*diff(dp(x),x)+b*dp(x)+c*p( x)=f(x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$eq1G/,(*&%\"aG\"\"\",** &-%%diffG6$-&%\"uG6#F)6#%\"xGF4F)-F-6$-&%\"hGF2F3F4F)F)*&F/F)-F-6$F7-% \"$G6$F4\"\"#F)F)*&-F-6$-&F16#F@F3F4F)-F-6$-&F9FFF3F4F)F)*&FDF)-F-6$FI F=F)F)F)F)*&%\"bGF),&*&F/F)F5F)F)*&FDF)FGF)F)F)F)*&%\"cGF),&*&F/F)F7F) F)*&FDF)FIF)F)F)F)-%\"fGF3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "In order to perform some algebraic manipulation s, substitute " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "di ff(h[1](x),`$`(x,2)) = m[1](x),diff(h[2](x),`$`(x,2)) = m[2](x),diff(h [1](x),x) = k[1](x),diff(h[2](x),x) = k[2](x);" "6&/-%%diffG6$-&%\"hG6 #\"\"\"6#%\"xG-%\"$G6$F-\"\"#-&%\"mG6#F+6#F-/-F%6$-&F)6#F16#F--F/6$F-F 1-&F46#F16#F-/-F%6$-&F)6#F+6#F-F--&%\"kG6#F+6#F-/-F%6$-&F)6#F16#F-F--& FM6#F16#F-" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 189 "subs(\{diff(h[1](x),x$2)=m[ 1](x),diff(h[2](x),x$2)=m[2](x)\},eq1):\nsubs(\{diff(h[1](x),x)=k[1](x ),diff(h[2](x),x)=k[2](x)\},%):\neq2 := subs(\{diff(u[1](x),x)=v[1](x) ,diff(u[2](x),x)=v[2](x)\},%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$e q2G/,(*&%\"aG\"\"\",**&-&%\"vG6#F)6#%\"xGF)-&%\"kGF/F0F)F)*&-&%\"uGF/F 0F)-&%\"mGF/F0F)F)*&-&F.6#\"\"#F0F)-&F4F?F0F)F)*&-&F8F?F0F)-&F;F?F0F)F )F)F)*&%\"bGF),&*&F6F)F2F)F)*&FDF)FAF)F)F)F)*&%\"cGF),&*&F6F)-&%\"hGF/ F0F)F)*&FDF)-&FSF?F0F)F)F)F)-%\"fGF0" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "Since we assumed that " }{XPPEDIT 18 0 "h[1](x);" "6#-&%\"hG6#\"\"\"6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "h[2](x);" "6#-&%\"hG6#\"\"#6#%\"xG" }{TEXT -1 100 " satisfy the \+ associated homogeneous equation we can simplify this expression by usi ng the facts that" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "a*m[1](x)+b*k[1](x)+c*h[1](x) = 0;" "6#/,(*&%\"aG\"\"\"-&%\"mG6#F'6 #%\"xGF'F'*&%\"bGF'-&%\"kG6#F'6#F-F'F'*&%\"cGF'-&%\"hG6#F'6#F-F'F'\"\" !" }}{PARA 0 "" 0 "" {TEXT -1 7 " and" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "a*m[2](x)+b*k[2](x)+c*h[2](x) = 0;" "6#/,(* &%\"aG\"\"\"-&%\"mG6#\"\"#6#%\"xGF'F'*&%\"bGF'-&%\"kG6#F,6#F.F'F'*&%\" cGF'-&%\"hG6#F,6#F.F'F'\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "expand(eq2);\nalg subs(a*m[1](x)+b*k[1](x)+c*h[1](x)=0,%):\nalgsubs(a*m[2](x)+b*k[2](x)+ c*h[2](x)=0,%);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2*( %\"aG\"\"\"-&%\"vG6#F'6#%\"xGF'-&%\"kGF+F,F'F'*(F&F'-&%\"uGF+F,F'-&%\" mGF+F,F'F'*(F&F'-&F*6#\"\"#F,F'-&F0F;F,F'F'*(F&F'-&F4F;F,F'-&F7F;F,F'F '*(%\"bGF'F2F'F.F'F'*(FEF'F@F'F=F'F'*(%\"cGF'F2F'-&%\"hGF+F,F'F'*(FHF' F@F'-&FKF;F,F'F'-%\"fGF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&*&- &%\"vG6#\"\"#6#%\"xG\"\"\"-&%\"kGF+F-F/F/*&-&%\"uGF+F-F/-&%\"mGF+F-F/F /F/%\"aGF/F/*(F:F/-&F*6#F/F-F/-&F2F>F-F/F/*(F:F/F4F/F7F/!\"\"-%\"fGF- " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*(%\"aG\"\"\"-&%\"vG6#\"\"#6#% \"xGF'-&%\"kGF+F-F'F'*(F&F'-&F*6#F'F-F'-&F1F5F-F'F'-%\"fGF-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Thus we end up with the second \+ relationship" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "a*v[1](x)*k[1](x)+a*v[2](x)*k[2](x) \+ = f(x);" "6#/,&*(%\"aG\"\"\"-&%\"vG6#F'6#%\"xGF'-&%\"kG6#F'6#F-F'F'*(F &F'-&F*6#\"\"#6#F-F'-&F06#F76#F-F'F'-%\"fG6#F-" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "v[1](x)*k[1](x)+v[2](x)*k[2](x) = f(x)/a;" "6#/,&*&-&% \"vG6#\"\"\"6#%\"xGF*-&%\"kG6#F*6#F,F*F**&-&F(6#\"\"#6#F,F*-&F/6#F66#F ,F*F**&-%\"fG6#F,F*%\"aG!\"\"" }{TEXT -1 14 " ------- (B)" }}{PARA 0 "" 0 "" {TEXT -1 37 "We can solve the system of equations:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 7 " " } {XPPEDIT 18 0 "v[1](x)*h[1](x)+v[2](x)*h[2](x) = 0;" "6#/,&*&-&%\"vG6# \"\"\"6#%\"xGF*-&%\"hG6#F*6#F,F*F**&-&F(6#\"\"#6#F,F*-&F/6#F66#F,F*F* \"\"!" }{TEXT -1 16 " ------- (A)" }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "v[1](x)*k[1](x)+v[2](x)*k[2](x) = f(x)/a;" "6#/ ,&*&-&%\"vG6#\"\"\"6#%\"xGF*-&%\"kG6#F*6#F,F*F**&-&F(6#\"\"#6#F,F*-&F/ 6#F66#F,F*F**&-%\"fG6#F,F*%\"aG!\"\"" }{TEXT -1 13 " ------- (B)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "by Cramer 's rule for " }{XPPEDIT 18 0 "v[1](x);" "6#-&%\"vG6#\"\"\"6#%\"xG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "v[2](x);" "6#-&%\"vG6#\"\"#6#%\"xG " }{TEXT -1 31 ", which are the derivatives of " }{XPPEDIT 18 0 "u[1]( x);" "6#-&%\"uG6#\"\"\"6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u[ 2](x);" "6#-&%\"uG6#\"\"#6#%\"xG" }{TEXT -1 15 " respectively. " }} {PARA 0 "" 0 "" {TEXT -1 5 " Let " }{XPPEDIT 18 0 "w(x) = ``;" "6#/-% \"wG6#%\"xG%!G" }{TEXT -1 5 " det " }{XPPEDIT 18 0 "matrix([[h[1](x), \+ h[2](x)], [k[1](x), k[2](x)]]);" "6#-%'matrixG6#7$7$-&%\"hG6#\"\"\"6#% \"xG-&F*6#\"\"#6#F.7$-&%\"kG6#F,6#F.-&F76#F26#F." }{TEXT -1 15 ", whic h is the " }{TEXT 259 9 "Wronskian" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "h[1](x);" "6#-&%\"hG6#\"\"\"6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "h[2](x);" "6#-&%\"hG6#\"\"#6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Then " }{XPPEDIT 18 0 "v[1](x) = -h[2](x)*f(x)/(a*w(x));" "6#/-&%\"vG6#\"\"\"6#%\"xG,$* (-&%\"hG6#\"\"#6#F*F(-%\"fG6#F*F(*&%\"aGF(-%\"wG6#F*F(!\"\"F;" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "v[2](x) = h[1](x)*f(x)/(a*w(x));" "6#/- &%\"vG6#\"\"#6#%\"xG*(-&%\"hG6#\"\"\"6#F*F0-%\"fG6#F*F0*&%\"aGF0-%\"wG 6#F*F0!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "We can then attempt to compute " }{XPPEDIT 18 0 "u[1](x);" "6#-&%\"uG6#\"\"\"6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u[2](x);" "6#-&%\"uG6#\"\"#6#%\"xG" }{TEXT -1 38 " by means of t he following integrals. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "u[1](x) = Int(v[1](x),x);" "6#/-&%\"uG6#\"\"\"6#%\"xG-% $IntG6$-&%\"vG6#F(6#F*F*" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "u[2]( x) = Int(v[2](x),x);" "6#/-&%\"uG6#\"\"#6#%\"xG-%$IntG6$-&%\"vG6#F(6#F *F*" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 263 24 "________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "Variation of parameters: summary of metho d" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 83 "Suppose that the 2nd order linear differential equation w ith 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 13 " ------- (N)" }}{PARA 0 "" 0 "" {TEXT -1 8 "has the " }{TEXT 259 22 "complementary solution" } {TEXT -1 1 " " }{XPPEDIT 18 0 "c(x) = C[1]*h[1](x)+C[2]*h[2](x);" "6#/ -%\"cG6#%\"xG,&*&&%\"CG6#\"\"\"F--&%\"hG6#F-6#F'F-F-*&&F+6#\"\"#F--&F0 6#F66#F'F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "d/dx;" "6#*&%\"dG\" \"\"%#dxG!\"\"" }{XPPEDIT 18 0 "[h[1](x)] = k[1](x);" "6#/7#-&%\"hG6# \"\"\"6#%\"xG-&%\"kG6#F)6#F+" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "d/dx; " "6#*&%\"dG\"\"\"%#dxG!\"\"" }{XPPEDIT 18 0 "[h[2](x)] = k[2](x);" "6 #/7#-&%\"hG6#\"\"#6#%\"xG-&%\"kG6#F)6#F+" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "w(x) =``" "6#/-%\"wG6#%\"xG%!G" }{TEXT -1 5 " det " } {XPPEDIT 18 0 "matrix([[h[1](x), h[2](x)], [k[1](x), k[2](x)]]);" "6#- %'matrixG6#7$7$-&%\"hG6#\"\"\"6#%\"xG-&F*6#\"\"#6#F.7$-&%\"kG6#F,6#F.- &F76#F26#F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Form " } {XPPEDIT 18 0 "v[1](x) = -h[2](x)*f(x)/(a*w(x));" "6#/-&%\"vG6#\"\"\"6 #%\"xG,$*(-&%\"hG6#\"\"#6#F*F(-%\"fG6#F*F(*&%\"aGF(-%\"wG6#F*F(!\"\"F; " }{TEXT -1 7 " and " }{XPPEDIT 18 0 "v[2](x) = h[1](x)*f(x)/(a*w(x) );" "6#/-&%\"vG6#\"\"#6#%\"xG*(-&%\"hG6#\"\"\"6#F*F0-%\"fG6#F*F0*&%\"a GF0-%\"wG6#F*F0!\"\"" }{TEXT -1 31 " , and determine (if possible):" } }{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "u[1](x) = Int(v[1]( x),x);" "6#/-&%\"uG6#\"\"\"6#%\"xG-%$IntG6$-&%\"vG6#F(6#F*F*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "u[2](x) = Int(v[2](x),x);" "6#/-&%\"uG6 #\"\"#6#%\"xG-%$IntG6$-&%\"vG6#F(6#F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Then a " }{TEXT 259 19 "particular integral" }{TEXT -1 12 " for (N) is " }{XPPEDIT 18 0 "p (x) = u[1](x)*h[1](x)+u[2](x)*h[2](x);" "6#/-%\"pG6#%\"xG,&*&-&%\"uG6# \"\"\"6#F'F.-&%\"hG6#F.6#F'F.F.*&-&F,6#\"\"#6#F'F.-&F26#F96#F'F.F." } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 85 ": This method can be applied in more \+ general situations than the one considered here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "Variation of par ameters: examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 5 "Solve" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)-4;" "6#,&*( %\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(\"\"%F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+4*y = (x+1)*exp(2*x);" "6#/,&*&%#dyG\"\"\"%#dxG! \"\"F'*&\"\"%F'%\"yGF'F'*&,&%\"xGF'F'F'F'-%$expG6#*&\"\"#F'F/F'F'" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "The auxiliary equation " }{XPPEDIT 18 0 "m^2-4*m+4 = 0;" "6#/,(*$%\"mG\"\"#\"\"\"*&\"\"%F(F& F(!\"\"F*F(\"\"!" }{TEXT -1 25 ", gives the single value " }{XPPEDIT 18 0 "m = 2" "6#/%\"mG\"\"#" }{TEXT -1 38 ", and so the complementary \+ solution is" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "c(x) = C[1]*exp(2*x)+C[2]*x*exp(2*x);" "6#/-%\"cG6#%\"xG,&*&&%\"CG6#\"\"\" F--%$expG6#*&\"\"#F-F'F-F-F-*(&F+6#F2F-F'F--F/6#*&F2F-F'F-F-F-" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "Thus we take " } {XPPEDIT 18 0 "h[1](x) = exp(2*x);" "6#/-&%\"hG6#\"\"\"6#%\"xG-%$expG6 #*&\"\"#F(F*F(" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "h[2](x) = x*exp(2* x);" "6#/-&%\"hG6#\"\"#6#%\"xG*&F*\"\"\"-%$expG6#*&F(F,F*F,F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[h[1](x)];" "6#7#-&%\"hG6#\"\"\"6#%\"xG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "k[1](x) = 2*exp(2*x);" "6#/-&%\"kG6#\"\"\"6 #%\"xG*&\"\"#F(-%$expG6#*&F,F(F*F(F(" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "d/dx;" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[h[2](x)];" "6#7#-&%\"hG6#\"\"#6#%\"xG" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "k[2](x) = (2*x+1)*exp(2*x);" "6#/-&%\"kG6#\"\"#6#%\"xG* &,&*&F(\"\"\"F*F.F.F.F.F.-%$expG6#*&F(F.F*F.F." }{TEXT -1 6 ", so " } }{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "w(x) = ``;" "6#/-% \"wG6#%\"xG%!G" }{TEXT -1 4 "det " }{XPPEDIT 18 0 "matrix([[h[1](x), h [2](x)], [k[1](x), k[2](x)]]) = ``;" "6#/-%'matrixG6#7$7$-&%\"hG6#\"\" \"6#%\"xG-&F+6#\"\"#6#F/7$-&%\"kG6#F-6#F/-&F86#F36#F/%!G" }{TEXT -1 4 "det " }{XPPEDIT 18 0 "matrix([[exp(2*x), x*exp(2*x)], [2*exp(2*x), (2 *x+1)*exp(2*x)]]) = exp(4*x);" "6#/-%'matrixG6#7$7$-%$expG6#*&\"\"#\" \"\"%\"xGF.*&F/F.-F*6#*&F-F.F/F.F.7$*&F-F.-F*6#*&F-F.F/F.F.*&,&*&F-F.F /F.F.F.F.F.-F*6#*&F-F.F/F.F.-F*6#*&\"\"%F.F/F." }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "h[1] := x -> exp(2*x);\nh[2 ] := x -> x*exp(2*x);\nwith(linalg):\nwronskian([h[1](x),h[2](x)],x); \ndet(%);\nw := unapply(simplify(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"hG6#\"\"\"f*6#%\"xG6\"6$%)operatorG%&arrowGF+-%$expG6#,$*& \"\"#F'9$F'F'F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"hG6#\"\"#f *6#%\"xG6\"6$%)operatorG%&arrowGF+*&9$\"\"\"-%$expG6#,$*&F'F1F0F1F1F1F +F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$-%$expG6#,$*& \"\"#\"\"\"%\"xGF.F.*&F/F.F(F.7$,$*&F-F.F(F.F.,&F(F.*(F-F.F/F.F(F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)-%$expG6#,$*&\"\"#\"\"\"%\"xGF+F+ F*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wGf*6#%\"xG6\"6$%)operator G%&arrowGF(-%$expG6#,$*&\"\"%\"\"\"9$F2F2F(F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v[1](x) = -h[2](x)*f(x)/(a* w(x));" "6#/-&%\"vG6#\"\"\"6#%\"xG,$*(-&%\"hG6#\"\"#6#F*F(-%\"fG6#F*F( *&%\"aGF(-%\"wG6#F*F(!\"\"F;" }{XPPEDIT 18 0 "`` = x*exp(2*x)*(x+1)*ex p(2*x)/exp(4*x);" "6#/%!G*,%\"xG\"\"\"-%$expG6#*&\"\"#F'F&F'F',&F&F'F' F'F'-F)6#*&F,F'F&F'F'-F)6#*&\"\"%F'F&F'!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 " ``= -x^2-x" "6#/%!G,&*$%\"xG\"\"#!\"\"F'F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v[2](x) = h[1 ](x)*f(x)/(a*w(x));" "6#/-&%\"vG6#\"\"#6#%\"xG*(-&%\"hG6#\"\"\"6#F*F0- %\"fG6#F*F0*&%\"aGF0-%\"wG6#F*F0!\"\"" }{XPPEDIT 18 0 "`` = exp(2*x)*( x+1)*exp(2*x)/exp(4*x);" "6#/%!G**-%$expG6#*&\"\"#\"\"\"%\"xGF+F+,&F,F +F+F+F+-F'6#*&F*F+F,F+F+-F'6#*&\"\"%F+F,F+!\"\"" }{XPPEDIT 18 0 " ``= \+ x+1" "6#/%!G,&%\"xG\"\"\"F'F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u [1](x) = Int(-x^2-x,x);" "6#/-&%\"uG6#\"\"\"6#%\"xG-%$IntG6$,&*$F*\"\" #!\"\"F*F1F*" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "-x^3/3-x^2/2;" "6#,&*& %\"xG\"\"$F&!\"\"F'*&F%\"\"#F)F'F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[ 2](x) = Int(x+1,x);" "6#/-&%\"uG6#\"\"#6#%\"xG-%$IntG6$,&F*\"\"\"F/F/F *" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "x^2/2+x;" "6#,&*&%\"xG\"\"#F&!\" \"\"\"\"F%F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }{XPPEDIT 18 0 "P(x) = (-x^3/3-x^2/2)*exp(2*x)+(x^2/2+x)*x*exp(2*x);" "6#/-%\"PG 6#%\"xG,&*&,&*&F'\"\"$F,!\"\"F-*&F'\"\"#F/F-F-\"\"\"-%$expG6#*&F/F0F'F 0F0F0*(,&*&F'F/F/F-F0F'F0F0F'F0-F26#*&F/F0F'F0F0F0" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "(x^3/6+x^2/2)*exp(2*x);" "6#*&,&*&%\"xG\"\"$\"\"'!\"\" \"\"\"*&F&\"\"#F,F)F*F*-%$expG6#*&F,F*F&F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "We can perform \+ the last few steps with Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 276 "f := x -> (x+1)*exp(2*x);\n -h[2](x)*f(x)/w(x);\nsimplify(%);\nv[1] := unapply(%,x);\nh[1](x)*f(x) /w(x);\nsimplify(%);\nv[2] := unapply(%,x);\nInt(v[1](x),x);\nvalue(%) ;\nu[1] := unapply(%,x);\nInt(v[2](x),x);\nvalue(%);\nu[2] := unapply( %,x);\nu[1](x)*h[1](x)+u[2](x)*h[2](x);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,&9$\" \"\"F/F/F/-%$expG6#,$*&\"\"#F/F.F/F/F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**%\"xG\"\"\"-%$expG6#,$*&\"\"#F&F%F&F&F,,&F%F&F&F&F& -F(6#,$*&\"\"%F&F%F&F&!\"\"F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&% \"xG\"\"\",&F%F&F&F&F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"vG 6#\"\"\"f*6#%\"xG6\"6$%)operatorG%&arrowGF+,$*&9$F',&F1F'F'F'F'!\"\"F+ F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%$expG6#,$*&\"\"#\"\"\"%\"x GF*F*F),&F+F*F*F*F*-F%6#,$*&\"\"%F*F+F*F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"\"F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>& %\"vG6#\"\"#f*6#%\"xG6\"6$%)operatorG%&arrowGF+,&9$\"\"\"F1F1F+F+F+" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&%\"xG\"\"\",&F(F)F)F)F)! \"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"$F&*$)%\"xGF' F&F&!\"\"*&#F&\"\"#F&*$)F*F.F&F&F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%\"uG6#\"\"\"f*6#%\"xG6\"6$%)operatorG%&arrowGF+,&*&#F'\"\"$F'*$)9$F 2F'F'!\"\"*&#F'\"\"#F'*$)F5F9F'F'F6F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&%\"xG\"\"\"F(F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"#F&*$)%\"xGF'F&F&F&F*F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"#f*6#%\"xG6\"6$%)operatorG%&arrowGF+, &*&#\"\"\"F'F2*$)9$F'F2F2F2F5F2F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&*&#\"\"\"\"\"$F(*$)%\"xGF)F(F(!\"\"*&#F(\"\"#F(*$)F,F0F(F(F-F (-%$expG6#,$*&F0F(F,F(F(F(F(*(,&*&#F(F0F(F1F(F(F,F(F(F,F(F3F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"'F&*(-%$expG6#,$*&\"\"# F&%\"xGF&F&F&)F/F.F&,&F/F&\"\"$F&F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The general solution of is " }{XPPEDIT 18 0 "y(x) = c(x)+ p(x);" "6#/-%\"yG6#%\"xG,&-%\"cG6#F'\"\"\"-%\"pG6#F'F," }{TEXT -1 10 " , that is," }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y(x) = C[1](x)*exp(2*x)+C[2]*x*exp(2*x)+(x^3/6+x^2/2)*exp(2*x);" "6#/-%\"yG6 #%\"xG,(*&-&%\"CG6#\"\"\"6#F'F.-%$expG6#*&\"\"#F.F'F.F.F.*(&F,6#F4F.F' F.-F16#*&F4F.F'F.F.F.*&,&*&F'\"\"$\"\"'!\"\"F.*&F'F4F4F@F.F.-F16#*&F4F .F'F.F.F." }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "We can check the solution by letting " }{TEXT 0 6 "dsolve" }{TEXT -1 18 " do the whole job." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "de := diff(y(x),x$ 2)-4*diff(y(x),x)+4*y(x)=(x+1)*exp(2*x);\ndsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\" \"#\"\"\"*&\"\"%F2-F(6$F*F-F2!\"\"*&F4F2F*F2F2*&,&F-F2F2F2F2-%$expG6#, $*&F1F2F-F2F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&- %$expG6#,$*&\"\"#\"\"\"F'F0F0F0%$_C2GF0F0*(F'F0F*F0%$_C1GF0F0*&#F0\"\" 'F0*&,(*$)F'\"\"$F0F0*&F;F0)F'F/F0F0*&F/F0F'F0F0F0F*F0F0F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Note that the t erm " }{XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "x*exp(2*x)" "6#*&%\"xG\"\"\"-%$expG6#*&\"\"#F%F$F%F%" } {TEXT -1 31 " can be absorbed into the term " }{XPPEDIT 18 0 "_C1*x*ex p(2*x);" "6#*(%$_C1G\"\"\"%\"xGF%-%$expG6#*&\"\"#F%F&F%F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 264 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 5 " Solve" }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "d^2*y/(d* x^2)+9*y = sec(3*x);" "6#/,&*(%\"dG\"\"#%\"yG\"\"\"*&F&F)*$%\"xGF'F)! \"\"F)*&\"\"*F)F(F)F)-%$secG6#*&\"\"$F)F,F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 103 "showing all the steps when the method of varia tion of parameters is used to find a particular integral." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 261 8 "Solution" } {TEXT 265 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 11 "We can use " }{TEXT 0 6 "dsolve" }{TEXT -1 36 " to find the complementary function." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "deH := diff(y(x),x$2)+9*y(x)=0;\ndsolve(deH,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$deHG/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\" \"\"*&\"\"*F2F*F2F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#% \"xG,&*&%$_C1G\"\"\"-%$sinG6#,$*&\"\"$F+F'F+F+F+F+*&%$_C2GF+-%$cosGF.F +F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "S et up functions " }{XPPEDIT 18 0 "h[1](x) = cos(3*x);" "6#/-&%\"hG6#\" \"\"6#%\"xG-%$cosG6#*&\"\"$F(F*F(" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "h[2](x) = sin(3*x);" "6#/-&%\"hG6#\"\"#6#%\"xG-%$sinG6#*&\"\"$\"\"\"F *F0" }{TEXT -1 24 " and find the Wronskian." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "h[1] := x -> cos( 3*x);\nh[2] := x -> sin(3*x);\nwith(linalg):\nwronskian([h[1](x),h[2]( x)],x);\ndet(%);\nw := unapply(simplify(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"hG6#\"\"\"f*6#%\"xG6\"6$%)operatorG%&arrowGF+-%$co sG6#,$*&\"\"$F'9$F'F'F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"hG 6#\"\"#f*6#%\"xG6\"6$%)operatorG%&arrowGF+-%$sinG6#,$*&\"\"$\"\"\"9$F5 F5F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$-%$cosG6#, $*&\"\"$\"\"\"%\"xGF.F.-%$sinGF*7$,$*&F-F.F0F.!\"\",$*&F-F.F(F.F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"$\"\"\")-%$cosG6#,$*&F%F&%\"xG F&F&\"\"#F&F&*&F%F&)-%$sinGF*F.F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"wG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 " Set up the fun ction " }{XPPEDIT 18 0 "f(x) = sec(3*x);" "6#/-%\"fG6#%\"xG-%$secG6#*& \"\"$\"\"\"F'F-" }{TEXT -1 61 " and complete the steps needed to find \+ a particular solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 270 "f := x -> sec(3*x);\n-h[2](x)*f(x) /w(x);\nsimplify(%);\nv[1] := unapply(%,x);\nh[1](x)*f(x)/w(x);\nsimpl ify(%);\nv[2] := unapply(%,x);\nInt(v[1](x),x);\nvalue(%);\nu[1] := un apply(%,x);\nInt(v[2](x),x);\nvalue(%);\nu[2] := unapply(%,x);\nu[1](x )*h[1](x)+u[2](x)*h[2](x);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$secG6#,$*&\"\"$\"\" \"9$F2F2F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"$F&*& -%$sinG6#,$*&F'F&%\"xGF&F&F&-%$secGF+F&F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"$F&*&-%$sinG6#,$*&F'F&%\"xGF&F&F&-%$cosG F+!\"\"F&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"vG6#\"\"\"f*6#%\"x G6\"6$%)operatorG%&arrowGF+,$*&#F'\"\"$F'*&-%$sinG6#,$*&F2F'9$F'F'F'-% $cosGF6!\"\"F'F&% \"vG6#\"\"##\"\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$ *&#\"\"\"\"\"$F)*&-%$sinG6#,$*&F*F)%\"xGF)F)F)-%$cosGF.!\"\"F)F4F1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"*F&-%#lnG6#-%$cosG6#,$* &\"\"$F&%\"xGF&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\" \"f*6#%\"xG6\"6$%)operatorG%&arrowGF+,$*&#F'\"\"*F'-%#lnG6#-%$cosG6#,$ *&\"\"$F'9$F'F'F'F'F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$ #\"\"\"\"\"$%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"$!\"\"%\" xG\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"#f*6#%\"xG6 \"6$%)operatorG%&arrowGF+,$*&#\"\"\"\"\"$F29$F2F2F+F+F+" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"*F&*&-%#lnG6#-%$cosG6#,$*&\"\"$F& %\"xGF&F&F&F,F&F&F&*&#F&F1F&*&F2F&-%$sinGF.F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"*F&*&-%#lnG6#-%$cosG6#,$*&\"\"$F&%\"xGF& F&F&F,F&F&F&*&#F&F1F&*&F2F&-%$sinGF.F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The general solution of is " } {XPPEDIT 18 0 "y(x) = c(x)+p(x);" "6#/-%\"yG6#%\"xG,&-%\"cG6#F'\"\"\"- %\"pG6#F'F," }{TEXT -1 10 ", that is," }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y(x) = C[1](x)*cos(3*x)+C[2]*sin(3*x)+ln(cos(3*x) )*cos(3*x)/9+x*sin(3*x)/3;" "6#/-%\"yG6#%\"xG,**&-&%\"CG6#\"\"\"6#F'F. -%$cosG6#*&\"\"$F.F'F.F.F.*&&F,6#\"\"#F.-%$sinG6#*&F4F.F'F.F.F.*(-%#ln G6#-F16#*&F4F.F'F.F.-F16#*&F4F.F'F.F.\"\"*!\"\"F.*(F'F.-F:6#*&F4F.F'F. F.F4FHF." }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 47 "Check by finding the solution completely using " } {TEXT 0 6 "dsolve" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "de := diff(y(x),x$2)+9*y(x)= sec(3*x);\ndsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/ ,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&\"\"*F2F*F2F2-%$secG6# ,$*&\"\"$F2F-F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,** &-%$sinG6#,$*&\"\"$\"\"\"F'F0F0F0%$_C2GF0F0*&-%$cosGF,F0%$_C1GF0F0*&#F 0\"\"*F0*&-%#lnG6#F3F0F3F0F0F0*&#F0F/F0*&F'F0F*F0F0F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 266 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 5 "Solve" }}{PARA 256 "" 0 "" {TEXT -1 4 " 4 " }{XPPEDIT 18 0 "d^2*y/(d*x^2)-4;" "6#,&*(%\"dG\" \"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(\"\"%F," }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx+y = exp(x/2)*sqrt(1-x^2);" "6#/,&*&%#dyG\"\"\"%#d xG!\"\"F'%\"yGF'*&-%$expG6#*&%\"xGF'\"\"#F)F'-%%sqrtG6#,&F'F'*$F0F1F)F '" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 103 "showing all the st eps when the method of variation of parameters is used to find a parti cular integral." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 261 8 "Solution" }{TEXT 267 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 11 "We can use " }{TEXT 0 6 "dsolve" }{TEXT -1 36 " to find the comple mentary function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "deH := 4*diff(y(x),x$2)-4*diff(y(x),x)+y(x)=0 ;\ndsolve(deH,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$deHG/,(*&\" \"%\"\"\"-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F0\"\"#F)F)*&F(F)-F+6$F-F0F)! \"\"F-F)\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&%$_ C1G\"\"\"-%$expG6#,$*&\"\"#!\"\"F'F+F+F+F+*(%$_C2GF+F,F+F'F+F+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Set up fu nctions " }{XPPEDIT 18 0 "h[1](x) = exp(x/2);" "6#/-&%\"hG6#\"\"\"6#% \"xG-%$expG6#*&F*F(\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "h[2 ](x) = x*exp(x/2);" "6#/-&%\"hG6#\"\"#6#%\"xG*&F*\"\"\"-%$expG6#*&F*F, F(!\"\"F," }{TEXT -1 24 " and find the Wronskian." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "h[1] := x - > exp(x/2);\nh[2] := x -> x*exp(x/2);\nwith(linalg):\nwronskian([h[1]( x),h[2](x)],x);\ndet(%);\nw := unapply(simplify(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"hG6#\"\"\"f*6#%\"xG6\"6$%)operatorG%&arrowGF+ -%$expG6#,$*&#F'\"\"#F'9$F'F'F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>&%\"hG6#\"\"#f*6#%\"xG6\"6$%)operatorG%&arrowGF+*&9$\"\"\"-%$expG6#, $*&#F1F'F1F0F1F1F1F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG 6#7$7$-%$expG6#,$*&\"\"#!\"\"%\"xG\"\"\"F0*&F/F0F(F07$,$*&#F0F-F0F(F0F 0,&F(F0*&F5F0F1F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)-%$expG6#,$* &\"\"#!\"\"%\"xG\"\"\"F-F*F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG %$expG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 " Set up the function " }{XPPEDIT 18 0 "f(x) = exp(x/2)*sqrt(1-x^2);" "6#/-%\"fG6#%\"xG*&-% $expG6#*&F'\"\"\"\"\"#!\"\"F--%%sqrtG6#,&F-F-*$F'F.F/F-" }{TEXT -1 61 " and complete the steps needed to find a particular solution." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 290 "f := x -> exp(x/2)*sqrt(1-x^2);\n-h[2](x)*f(x)/(4*w(x));\nsimplif y(%);\nv[1] := unapply(%,x);\nh[1](x)*f(x)/(4*w(x));\nsimplify(%);\nv[ 2] := unapply(%,x);\nInt(v[1](x),x);\nvalue(%);\nu[1] := unapply(%,x); \nInt(v[2](x),x);\nvalue(%);\nu[2] := unapply(%,x);\nu[1](x)*h[1](x)+u [2](x)*h[2](x);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"f Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&-%$expG6#,$*&#\"\"\"\"\"#F39$F3F 3F3-%%sqrtG6#,&F3F3*$)F5F4F3!\"\"F3F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"%F&**%\"xGF&-%$expG6#,$*&\"\"#!\"\"F)F&F &F/,&F&F&*$)F)F/F&F0#F&F/-F+6#F)F0F&F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"%!\"\"%\"xG\"\"\",&F(F(*$)F'\"\"#F(F&#F(F,F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"vG6#\"\"\"f*6#%\"xG6\"6$%)operatorG%&ar rowGF+,$*&#F'\"\"%F'*&9$F',&F'F'*$)F4\"\"#F'!\"\"#F'F8F'F9F+F+F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"%F&*(-%$expG6#,$*&\"\"# !\"\"%\"xGF&F&F.,&F&F&*$)F0F.F&F/#F&F.-F*6#F0F/F&F&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*&\"\"%!\"\",&\"\"\"F(*$)%\"xG\"\"#F(F&#F(F,F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"vG6#\"\"#f*6#%\"xG6\"6$%)operator G%&arrowGF+,$*&#\"\"\"\"\"%F2*$,&F2F2*$)9$F'F2!\"\"#F2F'F2F2F+F+F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*(\"\"%!\"\"%\"xG\"\"\",&F+ F+*$)F*\"\"#F+F)#F+F/F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"#7! \"\",&\"\"\"F(*$)%\"xG\"\"#F(F&#\"\"$F,F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"\"f*6#%\"xG6\"6$%)operatorG%&arrowGF+,$*&# F'\"#7F'*$),&F'F'*$)9$\"\"#F'!\"\"#\"\"$F9F'F'F'F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"\"%!\"\",&\"\"\"F+*$)%\"xG\"\"#F+F) #F+F/F+F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(\"\")!\"\"%\"xG\"\"\" ,&F(F(*$)F'\"\"#F(F&#F(F,F(*&#F(F%F(-%'arcsinG6#F'F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"#f*6#%\"xG6\"6$%)operatorG%&arrowGF+, &*&#\"\"\"\"\")F2*&9$F2,&F2F2*$)F5F'F2!\"\"#F2F'F2F2*&F1F2-%'arcsinG6# F5F2F2F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"#7F&*&),& F&F&*$)%\"xG\"\"#F&!\"\"#\"\"$F.F&-%$expG6#,$*&F.F/F-F&F&F&F&F&*(,&*( \"\")F/F-F&F*#F&F.F&*&#F&F:F&-%'arcsinG6#F-F&F&F&F-F&F2F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"#CF&*&-%$expG6#,$*&\"\"#!\"\"% \"xGF&F&F&,(*&F.F&,&F&F&*$)F0F.F&F/#F&F.F&*&F5F&F3F6F&*(\"\"$F&F0F&-%' arcsinG6#F0F&F&F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The general solution of is " }{XPPEDIT 18 0 "y(x) = \+ C(x)+P(x);" "6#/-%\"yG6#%\"xG,&-%\"CG6#F'\"\"\"-%\"PG6#F'F," }{TEXT -1 10 ", that is," }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y(x) = C[1](x)*exp(x/2)+C[2]*x*exp(x/2)+exp(x/2)*sqrt(1-x^2)/12+exp(x /2)*x^2*sqrt(1-x^2)/24+exp(x/2)*x*arcsin(x)/8;" "6#/-%\"yG6#%\"xG,,*&- &%\"CG6#\"\"\"6#F'F.-%$expG6#*&F'F.\"\"#!\"\"F.F.*(&F,6#F4F.F'F.-F16#* &F'F.F4F5F.F.*(-F16#*&F'F.F4F5F.-%%sqrtG6#,&F.F.*$F'F4F5F.\"#7F5F.**-F 16#*&F'F.F4F5F.*$F'F4F.-FA6#,&F.F.*$F'F4F5F.\"#CF5F.**-F16#*&F'F.F4F5F .F'F.-%'arcsinG6#F'F.\"\")F5F." }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Check by finding the solu tion completely using " }{TEXT 0 6 "dsolve" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "de : = 4*diff(y(x),x$2)-4*diff(y(x),x)+y(x)=exp(x/2)*sqrt(1-x^2);\ndsolve(d e,y(x)):\nexpand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*&\" \"%\"\"\"-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F0\"\"#F)F)*&F(F)-F+6$F-F0F)! \"\"F-F)*&-%$expG6#,$*&F4F8F0F)F)F),&F)F)*$)F0F4F)F8#F)F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,,*&-%$expG6#,$*&\"\"#!\"\"F'\"\" \"F1F1%$_C2GF1F1*(F'F1F*F1%$_C1GF1F1*&#F1\"#7F1*&F*F1,&F1F1*$)F'F/F1F0 #F1F/F1F1*&#F1\"#CF1*(F;F1F*F1F9F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "T asks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 147 "Solve the following differential equations, showing al l the steps when the method of variation of parameters is used to find a particular integral." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }} {PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+3;" "6# ,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(\"\"$F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+2*y = sin(exp(x));" "6#/,&*&%#dyG\"\"\"%#dxG !\"\"F'*&\"\"#F'%\"yGF'F'-%$sinG6#-%$expG6#%\"xG" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)-3;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\" \"F(\"\"$F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+2*y = exp(3*x)/(1+e xp(x));" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"#F'%\"yGF'F'*&-%$expG6#* &\"\"$F'%\"xGF'F',&F'F'-F/6#F3F'F)" }{TEXT -1 2 " " }}{PARA 0 "" 0 " " {TEXT -1 35 "___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }} {PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)-2;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(F&F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+2*y = exp(x)*sec(x);" "6#/,&*&%#dyG\"\"\"%#d xG!\"\"F'*&\"\"#F'%\"yGF'F'*&-%$expG6#%\"xGF'-%$secG6#F1F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "_________________________________ __" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+2;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(! \"\"F(F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+y = exp(-x)*ln(x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'%\"yGF'*&-%$expG6#,$%\"xGF)F'-%#lnG6#F0 F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "___________________ ________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "_____________________________ ______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }