{VERSION 6 0 "IBM INTEL LINUX" "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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {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 "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 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 13 "# SECTION 5.2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "# Examp le 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "#Check syntax for differential operator D: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "? D " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "interface(imaginaryunit=i); # Frees the res erved name I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "I := f -> \+ f; # Define the identity operator" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"IGj+6#%\"fG6\"6$%)operatorG%&arrowGF(9$F(F(F(6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "(D+I)(f); # Check to see if operato r (D + I) works as expected" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%\"D G6#%\"fG\"\"\"F'F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "3*I(f ); # Check to see if operator (3*I) works as expected" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"$\"\"\"%\"fGF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "(D+I)((D+3*I)(f)); # This is the operator (D+1)( D+3) applied to f" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(--%#@@G6$%\"DG \"\"#6#%\"fG\"\"\"*&\"\"%F,-F(F*F,F,*&\"\"$F,F+F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "# Example 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "(D+3*t)(D(y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, &--%#@@G6$%\"DG\"\"#6#%\"yG\"\"\"*&\"\"$F,-%\"tG6#-F(F*F,F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "D((D+3*t)(y));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&--%#@@G6$%\"DG\"\"#6#%\"yG\"\"\"*&\"\"$F,-F(6# -%\"tGF*F,F," }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "subs(D(t(y)) = t*D(y)+y,%); # Produ ct rule for derivative w.r.t. t of t*y(t)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(--%#@@G6$%\"DG\"\"#6#%\"yG\"\"\"*(\"\"$F,%\"tGF,-F(F* F,F,*&F.F,F+F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "# Exa mple 3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "# Since operators are applied to \"Maple functions\", express th e nonhomogeneous" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "# terms as functions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f1 := t - > 1; f2 := t -> 10*t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1Gj+6#%\" tG6\"6$%)operatorG%&arrowGF(\"\"\"F(F(F(6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2Gj+6#%\"tG6\"6$%)operatorG%&arrowGF(,$*&\"#5\"\"\" 9$F/F/F(F(F(6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "de1 \+ := (D-3*I)(x) + 4*y = f1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de1G/, (-%\"DG6#%\"xG\"\"\"*&\"\"$F+F*F+!\"\"*&\"\"%F+%\"yGF+F+%#f1G" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "de2 := -4*x+(D+7*I)(y) = f2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de2G/,(*&\"\"%\"\"\"%\"xGF)!\" \"-%\"DG6#%\"yGF)*&\"\"(F)F/F)F)%#f2G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "4*de1 + (D-3*I)(de2); # Eliminate x" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&\"\"&\"\"\"%\"yGF'!\"\"--%#@@G6$%\"DG\"\"#6#F(F '*&\"\"%F'-F.F0F'F',(*&F2F'%#f1GF'F'\"#5F'*&\"\"$F'%#f2GF'F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "# Evaluate above differentia l equation at t:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "de := % (t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*&\"\"&\"\"\"-%\"yG6# %\"tGF)!\"\"---%#@@G6$%\"DG\"\"#6#F+F,F)*&\"\"%F)--F4F6F,F)F),&\"#9F)* &\"#IF)F-F)F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "# This is \+ Eq (3) in text, ehen the left hand side is evaluated at t." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "# Solve above differential equation by the method of undetermined coefficients:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "P \+ := r^2+4*r-5; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG,(*$)%\"rG\"\" #\"\"\"F**&\"\"%F*F(F*F*\"\"&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "rsol := solve(P=0,r); #Solve characteristic equation " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rsolG6$\"\"\"!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "r1 := rsol[2]; r2 := rsol[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#r1G!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2G\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "y1 := exp(r1*t); y2 := exp(r2*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G-%$expG6#,$*&\"\"&\"\"\"%\"tGF +!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2G-%$expG6#%\"tG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "yh := C1*y1+C2*y2; # Homogen eous solution" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yhG,&*&%#C1G\"\"\" -%$expG6#,$*&\"\"&F(%\"tGF(!\"\"F(F(*&%#C2GF(-F*6#F/F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "yp := A*t+B; # Assumed form of part icular solution" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ypG,&*&%\"AG\"\" \"%\"tGF(F(%\"BGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "de := diff(y(t),t,t) +4*diff(y(t),t) - 5*y(t) = 14-30*t; # Use diff instead of D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#% \"tG-%\"$G6$F-\"\"#\"\"\"*&\"\"%F2-F(6$F*F-F2F2*&\"\"&F2F*F2!\"\",&\"# 9F2*&\"#IF2F-F2F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(y (t)=yp,de);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*-%%diffG6$,&*&%\"AG \"\"\"%\"tGF+F+%\"BGF+-%\"$G6$F,\"\"#F+*&\"\"%F+-F&6$F(F,F+F+*(\"\"&F+ F*F+F,F+!\"\"*&F7F+F-F+F8,&\"#9F+*&\"#IF+F,F+F8" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /,(*&\"\"%\"\"\"%\"AGF'F'*(\"\"&F'F(F'%\"tGF'!\"\"*&F*F'%\"BGF'F,,&\"# 9F'*&\"#IF'F+F'F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eq := % ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/,(*&\"\"%\"\"\"%\"AGF)F)*( \"\"&F)F*F)%\"tGF)!\"\"*&F,F)%\"BGF)F.,&\"#9F)*&\"#IF)F-F)F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(t=0,%); " }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,$*&\"\"&\"\"\"%\"AGF'!\"\"!#S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "eq1 := %; # Equality of constant te rm (i.e., of coefficients of 1):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% $eq1G/,$*&\"\"&\"\"\"%\"AGF)!\"\"!#S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "diff(eq,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*& \"\"&\"\"\"%\"AGF'!\"\"!#S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "eq2 := %; # Equality of coefficients of t" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/,$*&\"\"&\"\"\"%\"AGF)!\"\"!#S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(\{eq1,eq2\},\{A,B\});" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"BGF%/%\"AG\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "subs(%,yp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\")\"\"\"%\"tGF&F&%\"BGF&" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 14 "ysol := yh+%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ysolG,**&%#C1G\"\"\"-%$expG6#,$*&\"\"&F(%\"tGF(!\"\"F(F(*&%#C 2GF(-F*6#F/F(F(*&\"\")F(F/F(F(%\"BGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "# Find x :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "# Method 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "(D+7*I)(de1) -4*de2; # Eliminate y " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(--%#@@G6$%\"DG\"\"#6#%\"xG\"\" \"*&\"\"%F--F)F+F-F-*&\"\"&F-F,F-!\"\",&*&\"\"(F-%#f1GF-F-*&F/F-%#f2GF -F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "%(t); # Evaluate at \+ t" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(---%#@@G6$%\"DG\"\"#6#%\"xG6#% \"tG\"\"\"*&\"\"%F0--F*F,F.F0F0*&\"\"&F0-F-F.F0!\"\",&\"\"(F0*&\"#SF0F /F0F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "# Solve above diff erential equation by the method of undetermined coefficients:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "# Homogeneous solution has s ame form as yh:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "xh := K1 *y1+K2*y2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xhG,&*&%#K1G\"\"\"-%$ expG6#,$*&\"\"&F(%\"tGF(!\"\"F(F(*&%#K2GF(-F*6#F/F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "xp := A*t+B; # Assumed form of part icular solution" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xpG,&*&%\"AG\"\" \"%\"tGF(F(%\"BGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "de := diff(x(t),t,t) +4*diff(x(t),t) - 5*x(t) = 7-40*t;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\" *&\"\"%F2-F(6$F*F-F2F2*&\"\"&F2F*F2!\"\",&\"\"(F2*&\"#SF2F-F2F9" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(x(t)=xp,de);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*-%%diffG6$,&*&%\"AG\"\"\"%\"tGF+F+%\"BGF+ -%\"$G6$F,\"\"#F+*&\"\"%F+-F&6$F(F,F+F+*(\"\"&F+F*F+F,F+!\"\"*&F7F+F-F +F8,&\"\"(F+*&\"#SF+F,F+F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eq := expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/,(*&\" \"%\"\"\"%\"AGF)F)*(\"\"&F)F*F)%\"tGF)!\"\"*&F,F)%\"BGF)F.,&\"\"(F)*& \"#SF)F-F)F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(t=0,eq );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&\"\"%\"\"\"%\"AGF'F'*&\"\"& F'%\"BGF'!\"\"\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eq1 \+ := %;# Equality of coefficients of 1" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%$eq1G/,&*&\"\"%\"\"\"%\"AGF)F)*&\"\"&F)%\"BGF)!\"\"\"\"(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "diff(eq,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&\"\"&\"\"\"%\"AGF'!\"\"!#S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "eq2 := % ;# Equality of coefficients of t" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/,$*&\"\"&\"\"\"%\"AGF)!\"\"!#S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(\{eq1,eq2\},\{A,B\});" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"AG\"\")/%\"BG\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "subs(%,xp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\")\"\"\"%\"tGF&F&\"\"&F&" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 15 "xsol := xh + %;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%xsolG,**&%#K1G\"\"\"-%$expG6#,$*&\"\"&F(%\"tGF(!\"\"F(F(*&%#K 2GF(-F*6#F/F(F(*&\"\")F(F/F(F(F.F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "# Rewrite eq1 in the form originally given, i.e., fir st of (1): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "de1new := di ff(x(t),t) = 3*x(t) - 4*y(t) + 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %'de1newG/-%%diffG6$-%\"xG6#%\"tGF,,(*&\"\"$\"\"\"F)F0F0*&\"\"%F0-%\"y GF+F0!\"\"F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "subs(\{x( t)=xsol,y(t)=ysol\},de1new);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%di ffG6$,**&%#K1G\"\"\"-%$expG6#,$*&\"\"&F*%\"tGF*!\"\"F*F**&%#K2GF*-F,6# F1F*F**&\"\")F*F1F*F*F0F*F1,,*(\"\"$F*F)F*F+F*F**(F;F*F4F*F5F*F*F8F**( \"\"%F*%#C1GF*F+F*F2*(F>F*%#C2GF*F5F*F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(\" \"&\"\"\"%#K1GF'-%$expG6#%\"tG!\"&!\"\"*&%#K2GF'F)F'F'\"\")F',,*(\"\"$ F'F(F'F)F-F'*(F4F'F0F'F)F'F'F1F'*(\"\"%F'%#C1GF'F)F-F.*(F7F'%#C2GF'F)F 'F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(\"\"&\"\"\"%#K1GF'-%$expG6#,$*&F& F'%\"tGF'!\"\"F'F/*&%#K2GF'-F*6#F.F'F'\"\")F',,*(\"\"$F'F(F'F)F'F'*(F7 F'F1F'F2F'F'F4F'*(\"\"%F'%#C1GF'F)F'F/*(F:F'%#C2GF'F2F'F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eq1 := -5*K1=3*K1 - 4*C1;# Coefficient of e^(-5t)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/,$*&\"\"&\"\"\"%#K1GF)!\"\",&*&\"\"$F)F*F)F)*& \"\"%F)%#C1GF)F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eq2 := \+ K2 = 3*K2-4*C2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/%#K2G,&*&\" \"$\"\"\"F&F*F**&\"\"%F*%#C2GF*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(\{eq1,eq2\},\{K1,K2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%#K2G,$*&\"\"#\"\"\"%#C2GF)F)/%#K1G,$*&F(!\"\"%#C1GF )F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs(%,[xsol,ysol]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,**&#\"\"\"\"\"#F'*&%#C1GF'-%$ex pG6#,$*&\"\"&F'%\"tGF'!\"\"F'F'F'*(F(F'%#C2GF'-F,6#F1F'F'*&\"\")F'F1F' F'F0F',*F)F'*&F4F'F5F'F'*&\"\"'F'F1F'F'F(F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "# Method 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "#Rewrite the second equation in (1) in the form" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "de2new := diff(y(t),t) = 4*x(t)-7*y (t) + 10*t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'de2newG/-%\"DG6#%\"y G,(*&\"\"%\"\"\"-%\"xG6#%\"tGF-F-*&\"\"(F--F)F0F-!\"\"*&\"#5F-F1F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(de2new,x(t));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"\"\"\"%F&-%\"DG6#%\"yGF&F&*&# \"\"(F'F&-F+6#%\"tGF&F&*(\"\"&F&\"\"#!\"\"F1F&F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(y(t)=ysol,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&#\"\"\"\"\"%F&-%%diffG6$,**&%#C1GF&-%$expG6#,$*&\" \"&F&%\"tGF&!\"\"F&F&*&%#C2GF&-F/6#F4F&F&*&\"\"'F&F4F&F&\"\"#F&F4F&F&* &#\"\"(F'F&F,F&F&*&F>F&F6F&F&*&\"\")F&F4F&F&#F?F " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\"\"\"\"#F&*&%#C1GF&-%$expG6#%\"tG!\"&F&F&*(F'F&%#C2GF&F* F&F&*&\"\")F&F-F&F&\"\"&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\"\"\"\"#F& *&%#C1GF&-%$expG6#,$*&\"\"&F&%\"tGF&!\"\"F&F&F&*(F'F&%#C2GF&-F+6#F0F&F &*&\"\")F&F0F&F&F/F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "xso l := %;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%xsolG,**&#\"\"\"\"\"#F(* &%#C1GF(-%$expG6#,$*&\"\"&F(%\"tGF(!\"\"F(F(F(*(F)F(%#C2GF(-F-6#F2F(F( *&\"\")F(F2F(F(F1F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "# Example 4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "# Check syntax for second derivative operator in term s of D:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "?D" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "# Express nonhomogeneous terms as f unctions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f1 := t -> -1; f2 := t -> t^2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1Gj+6#%\"tG6 \"6$%)operatorG%&arrowGF(!\"\"F(F(F(6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2Gj+6#%\"tG6\"6$%)operatorG%&arrowGF(*$)9$\"\"#\"\" \"F(F(F(6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "de1 := ( D@@2-I)(x) + (D+I)(y) = f1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de1G /,*--%#@@G6$%\"DG\"\"#6#%\"xG\"\"\"F.!\"\"-F+6#%\"yGF/F3F/%#f1G" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "de2 := (D-I)(x) + D(y) = f2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de2G/,(-%\"DG6#%\"xG\"\"\"F*!\" \"-F(6#%\"yGF+%#f2G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "L1 : = (D@@2-I); L2 := D+I; L3 := D-I; L4 := D; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#L1G,&-%#@@G6$%\"DG\"\"#\"\"\"%\"IG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#L2G,&%\"DG\"\"\"%\"IGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#L3G,&%\"DG\"\"\"%\"IG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#L4G%\"DG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "# Verify that these L's give de1 and de2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "de1new := L1(x) + L2(y) = f1;%(t);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'de1newG/,*--%#@@G6$%\"DG\"\"#6#%\"xG\"\"\"F.! \"\"-F+6#%\"yGF/F3F/%#f1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*---%#@ @G6$%\"DG\"\"#6#%\"xG6#%\"tG\"\"\"-F-F.!\"\"--F*6#%\"yGF.F0-F6F.F0F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "de2new := L3(x) + L4(y) = f2;%(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'de2newG/,(-%\"DG6#%\"x G\"\"\"F*!\"\"-F(6#%\"yGF+%#f2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,( --%\"DG6#%\"xG6#%\"tG\"\"\"-F)F*!\"\"--F'6#%\"yGF*F,*$)F+\"\"#F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "# These agree with de1 and d e2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "# Eliminating y:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "L4(de1new) - L2(de2new); " } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*--%#@@G6$%\"DG\"\"$6#%\"xG\"\"\"-F )F+!\"\"--F'6$F)\"\"#F+F/F,F-,&j+6#%\"tG6\"6$%)operatorG%&arrowGF8,$*& F3F-9$F-F-F8F8F86#\"+BfE8?F/%#f2GF/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "%(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*---%#@@G6$ %\"DG\"\"$6#%\"xG6#%\"tG\"\"\"--F*F,F.!\"\"---F(6$F*\"\"#F,F.F3-F-F.F0 ,&*&F8F0F/F0F3*$)F/F8F0F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "#To see that this is equivalent to Equation (11) in the text, we l ook at the " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "#charcterist ic polynomial of (11):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "P := (r-1)^2*(r+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG*&),&%\"rG \"\"\"F)!\"\"\"\"#F),&F(F)F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)% \"rG\"\"$\"\"\"F(*$)F&\"\"#F(!\"\"F&F,F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 29 "# This shows the equivalence." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "# Alternatively, suppose we started with P in the above form:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "P := %; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG,**$)%\"rG\"\"$\"\"\"F**$)F( \"\"#F*!\"\"F(F.F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "# T rying to find the roots by hand, we might look for small integer roots :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(r=1,P); " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "# This show r-1 is a factor:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "P/(r-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,** $)%\"rG\"\"$\"\"\"F)*$)F'\"\"#F)!\"\"F'F-F)F)F),&F'F)F)F-F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"rG\"\"#\"\"\"F(F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "# Shows P = (r-1)(r-1)(r+1)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "# Next we solve the homogeneous equ ation, and then use the method of undetermined # coefficients:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "xh := C1*exp(t) + C2*t*exp(t ) + C3*exp(-t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xhG,(*&%#C1G\"\" \"-%$expG6#%\"tGF(F(*(%#C2GF(F,F(F)F(F(*&%#C3GF(-F*6#,$F,!\"\"F(F(" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "xp := A*t^2+B*t+C;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xpG,(*&%\"AG\"\"\")%\"tG\"\"#F(F(*& %\"BGF(F*F(F(%\"CGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "de \+ := diff(x(t),t,t,t) - diff(x(t),t) -diff(x(t),t,t) + x(t) = -2*t-t^2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,*-%%diffG6$-%\"xG6#%\"tG-% \"$G6$F-\"\"$\"\"\"-F(6$F*F-!\"\"-F(6$F*-F/6$F-\"\"#F5F*F2,&*&F:F2F-F2 F5*$)F-F:F2F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(x(t)= xp,de);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,.-%%diffG6$,(*&%\"AG\"\" \")%\"tG\"\"#F+F+*&%\"BGF+F-F+F+%\"CGF+-%\"$G6$F-\"\"$F+-F&6$F(F-!\"\" -F&6$F(-F36$F-F.F8F)F+F/F+F1F+,&*&F.F+F-F+F8*$F,F+F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eq := expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/,.*(\"\"#\"\"\"%\"AGF)%\"tGF)!\"\"%\"BGF,*&F(F)F *F)F,*&F*F))F+F(F)F)*&F-F)F+F)F)%\"CGF),&*&F(F)F+F)F,*$F0F)F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(t=0,eq);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,(%\"BG!\"\"*&\"\"#\"\"\"%\"AGF)F&%\"CGF)\"\"! " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eq1 := %; # Coefficient of 1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/,(%\"BG!\"\"*&\"\"#\" \"\"%\"AGF+F(%\"CGF+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "diff(eq,t); subs(t=0,%); eq2 := %; # Coefficient of t " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&\"\"#\"\"\"%\"AGF'!\"\"*(F&F'F(F'%\"tGF 'F'%\"BGF',&F&F)*&F&F'F+F'F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*& \"\"#\"\"\"%\"AGF'!\"\"%\"BGF'!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%$eq2G/,&*&\"\"#\"\"\"%\"AGF)!\"\"%\"BGF)!\"#" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 46 "diff(eq,t,t)/2; eq3 := %; # Coefficient of t^2 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G/%\"AG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(\{eq1,eq2,eq3\},\{A,B,C\});" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<%/%\"AG!\"\"/%\"BG!\"%/%\"CG!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "subs(%,xp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"tG\"\"#\"\"\"!\"\"*&\"\"%F(F&F(F)\"\"'F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "xsol := xh + %;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%xsolG,.*&%#C1G\"\"\"-%$expG6#%\"tGF(F(*(%#C2GF(F,F (F)F(F(*&%#C3GF(-F*6#,$F,!\"\"F(F(*$)F,\"\"#F(F4*&\"\"%F(F,F(F4\"\"'F4 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "# Find y:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "de1 - de2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(--%#@@G6$%\"DG\"\"#6#%\"xG\"\"\"%\"yGF--F)F+!\"\",&% #f1GF-%#f2GF0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve(%,y) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*--%#@@G6$%\"DG\"\"#6#%\"xG!\"\" -F(F*\"\"\"%#f1GF.%#f2GF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "ysol := %(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ysolG,*---%#@@ G6$%\"DG\"\"#6#%\"xG6#%\"tG!\"\"--F+F-F/\"\"\"F4F1*$)F0F,F4F1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "# Rewriting and replacing x \+ by xsol gives\024" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "ysol : = - diff(xsol,t,t) + diff(xsol,t) -1 -t^2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ysolG,,*&%#C2G\"\"\"-%$expG6#%\"tGF(!\"\"*(\"\"#F(%# C3GF(-F*6#,$F,F-F(F-\"\"$F-*&F/F(F,F(F-*$)F,F/F(F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "# Example 5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "de1 := (D-I)(x) - 2*y +z = 0 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de1G/,*-%\"DG6#%\"xG\"\"\"F*! \"\"*&\"\"#F+%\"yGF+F,%\"zGF+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "de2 := -x + D(y)-z = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de2G/,(%\"xG!\"\"-%\"DG6#%\"yG\"\"\"%\"zGF(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "de3 := -4*x + 4*y + (D-5*I)(z) = 0; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de3G/,**&\"\"%\"\"\"%\"xGF)!\" \"*&F(F)%\"yGF)F)-%\"DG6#%\"zGF)*&\"\"&F)F1F)F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "# The above are Eq (14)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "# Eliminate z:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "de4 := de1 + de2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de4G/,*-%\"DG6#%\"xG\"\"\"*&\"\"#F+F*F+!\"\"*&F-F+%\"yGF+F.-F(6# F0F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "de5 := (D-5*I) (de2) + de3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de5G/,,-%\"DG6#%\"x G!\"\"--%#@@G6$F(\"\"#6#%\"yG\"\"\"F*F3*&\"\"&F3-F(F1F3F+*&\"\"%F3F2F3 F3\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "# de4 and de5 ar e Eq (16). Next, we eliminate x: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "de6 := (D-I)(de4)+(D-2*I)(de5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de6G/,**&\"#6\"\"\"-%\"DG6#%\"yGF)F)*&\"\"'F)--%#@@G 6$F+\"\"#F,F)!\"\"*&F/F)F-F)F5--F26$F+\"\"$F,F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "# Solve characteristic polynomial by look ing for small integer roots:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "P := 11*r-6*r^2-6+r^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG ,**&\"#6\"\"\"%\"rGF(F(*&\"\"'F()F)\"\"#F(!\"\"F+F.*$)F)\"\"$F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(r=1,P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "P/(r-1); simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,**&\"# 6\"\"\"%\"rGF'F'*&\"\"'F')F(\"\"#F'!\"\"F*F-*$)F(\"\"$F'F'F',&F(F'F'F- F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"rG\"\"#\"\"\"F(*&\"\"&F( F&F(!\"\"\"\"'F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"rG\"\"\"\"\"#!\"\"F&,&F% F&\"\"$F(F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "# Therefore \+ we have" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "P := (r-1)*(r-2) *(r-3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG*(,&%\"rG\"\"\"F(!\" \"F(,&F'F(\"\"#F)F(,&F'F(\"\"$F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ysol := C1*exp(t) + C2*exp(2*t) + C3*exp(3*t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ysolG,(*&%#C1G\"\"\"-%$expG6#%\"tGF (F(*&%#C2GF(-F*6#,$*&\"\"#F(F,F(F(F(F(*&%#C3GF(-F*6#,$*&\"\"$F(F,F(F(F (F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "# To find x, we add \+ de4 and de5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "de4 + de5; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*%\"xG!\"\"*&\"\"#\"\"\"%\"yGF)F )*&\"\"%F)-%\"DG6#F*F)F&--%#@@G6$F.F(F/F)\"\"!" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 11 "solve(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,(*&\"\"#\"\"\"%\"yGF&F&*&\"\"%F&-%\"DG6#F'F&!\"\"--%#@@G6$F+F%F,F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "xsol := 2*ysol - 4*diff(y sol,t) + diff(ysol,t,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%xsolG,( *&%#C1G\"\"\"-%$expG6#%\"tGF(!\"\"*(\"\"#F(%#C2GF(-F*6#,$*&F/F(F,F(F(F (F-*&%#C3GF(-F*6#,$*&\"\"$F(F,F(F(F(F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "# To find z, we use de2:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 4 "de2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(%\"xG!\" \"-%\"DG6#%\"yG\"\"\"%\"zGF&\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(de2,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&% \"xG!\"\"-%\"DG6#%\"yG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "zsol := -xsol + diff(ysol,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%%zsolG,(*(\"\"#\"\"\"%#C1GF(-%$expG6#%\"tGF(F(*(\"\"%F(%#C2GF(-F+6# ,$*&F'F(F-F(F(F(F(*(F/F(%#C3GF(-F+6#,$*&\"\"$F(F-F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "170 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }