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1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 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 48 "Introduction to partial different ial equations " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaim o, 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 13 "Introduction " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 130 "An equati on involving one or more partial derivatives of an (unknown) function \+ of two (or more) independent variables is called a " }{TEXT 259 29 "pa rtial differential equation" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "For example the partial differential equation: " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),`$`(x,2))" "6#-%%Dif fG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F)\"\"#" }{XPPEDIT 18 0 "``+``;" "6#,&%! G\"\"\"F$F%" }{XPPEDIT 18 0 "Diff(u(x,y),`$`(y,2)) = 0;" "6#/-%%DiffG6 $-%\"uG6$%\"xG%\"yG-%\"$G6$F+\"\"#\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "for the unknown func tion " }{XPPEDIT 18 0 "u(x,y)" "6#-%\"uG6$%\"xG%\"yG" }{TEXT -1 30 " o f the independent variables " }{TEXT 263 1 "x" }{TEXT -1 5 " and " } {TEXT 264 1 "y" }{TEXT -1 11 " is called " }{TEXT 259 18 "Laplace's eq uation" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 83 "A partial diff erential equation can have a wide range of very different solutions. \+ " }}{PARA 0 "" 0 "" {TEXT -1 13 "For example, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([u(x,y) = x^2-y^2, ``],[u(x,y ) = y^3-3*y*x^2, ``],[u(x,y) = y^4-6*y^2*x^2+x^4, ``],[u(x,y) =y^5-10* y^3*x^2+5*y*x^4 , ``],[u(x,y) = y^6-15*y^4*x^2+15*y^2*x^4-x^6, ``])" " 6#-%*PIECEWISEG6'7$/-%\"uG6$%\"xG%\"yG,&*$F+\"\"#\"\"\"*$F,F/!\"\"%!G7 $/-F)6$F+F,,&*$F,\"\"$F0*(F:F0F,F0F+F/F2F37$/-F)6$F+F,,(*$F,\"\"%F0*( \"\"'F0*$F,F/F0F+F/F2*$F+FBF0F37$/-F)6$F+F,,(*$F,\"\"&F0*(\"#5F0*$F,F: F0F+F/F2*(FMF0F,F0F+FBF0F37$/-F)6$F+F,,**$F,FDF0*(\"#:F0*$F,FBF0F+F/F2 *(FYF0*$F,F/F0F+FBF0*$F+FDF2F3" }{TEXT -1 8 " " }{XPPEDIT 18 0 "PIECEWISE([u(x,y) = exp(-y)*cos*x, ``],[u(x,y) = ln(x^2+y^2), ``],[u( x,y) = sin*y*cosh*x, ``],[u(x,y) = arctan(y/x), ``],[u(x,y) = exp(y^2- x^2)*cos*2*x*y, ``]);" "6#-%*PIECEWISEG6'7$/-%\"uG6$%\"xG%\"yG*(-%$exp G6#,$F,!\"\"\"\"\"%$cosGF3F+F3%!G7$/-F)6$F+F,-%#lnG6#,&*$F+\"\"#F3*$F, F?F3F57$/-F)6$F+F,**%$sinGF3F,F3%%coshGF3F+F3F57$/-F)6$F+F,-%'arctanG6 #*&F,F3F+F2F57$/-F)6$F+F,*,-F/6#,&*$F,F?F3*$F+F?F2F3F4F3F?F3F+F3F,F3F5 " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([u(x,y) = 2*cos*y*sin*y/(sin h^2*x+cos^2*y), ``],[u(x,y) = 2*cosh*x*sin*y/(cosh^2*x-cos^2*y), ``]) " "6#-%*PIECEWISEG6$7$/-%\"uG6$%\"xG%\"yG*.\"\"#\"\"\"%$cosGF/F,F/%$si nGF/F,F/,&*&%%sinhGF.F+F/F/*&F0F.F,F/F/!\"\"%!G7$/-F)6$F+F,*.F.F/%%cos hGF/F+F/F1F/F,F/,&*&F=F.F+F/F/*&F0F.F,F/F6F6F7" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "u(x,y)=2*cos(y^2)*cos(x^2)*cosh^2*``(y*x)-cos(y^2)*cos( x^2)+2*sin(y^2)*sin(x^2)*cosh^2*``(y*x)-sin(y^2)*sin(x^2)" "6#/-%\"uG6 $%\"xG%\"yG,**,\"\"#\"\"\"-%$cosG6#*$F(F+F,-F.6#*$F'F+F,%%coshGF+-%!G6 #*&F(F,F'F,F,F,*&-F.6#*$F(F+F,-F.6#*$F'F+F,!\"\"*,F+F,-%$sinG6#*$F(F+F ,-FC6#*$F'F+F,F4F+-F66#*&F(F,F'F,F,F,*&-FC6#*$F(F+F,-FC6#*$F'F+F,F@" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "are all solutions of Laplace's equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 137 "For a partial differe ntial equation corresponding to a physical system it is often possible to obtain a unique solution by making use of " }{TEXT 259 18 "initial conditions" }{TEXT -1 63 " determined by the initial state of the phy sical system and/or " }{TEXT 259 19 "boundary conditions" }{TEXT -1 46 " corresponding to fixed external constraints. 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1A678F4C01F3CAE0A71C80AC01AC86277D4C2F7C016F6B0E977EB1BC013B069AA68863 3C010E1AB6E495342C00CF2D53F0A6E34C008D213C6D36CA1C0054802BFCADE9EC0023 F25507AAA23BFFF4A1E0B6E6281BFFAD3E7F0DD4FAEBFF70091CD2FF209BFF3B8E1F20 8EEB2BFF0E8EE85BBD4BEBFECFF493B286748BFE8DCC12AD7A674BFE5512A5E7BF6FDB FE246FEBE45A563BFDF5793D43C4279BFDADF725EB2FE1FBFD70A76F0C8064FBFD3C15 DDE68F27BBFD0F034BCEC5643BFCD0BC292ADD839BFC8E77326B4C9BFBFC55A55ED5A2 3DDBFC24EDB8C7AA557BFBF650F6757FE1CBFBAEB01C3779E1DBFB7146056142321BFB 3C9DD7119D148BFB0F77E1532E019BFAD184147E8B414BFA8F229BC64A9B1BFA563856 E17177B-%&COLORG6&%$RGBG$\"\")!\"\"$\"\"'F;$\"\"\"F)-%+LIGHTMODELG6#%( LIGHT_4G-%+AXESLABELSG6%%\"xG%\"yG%\"uG-%*AXESSTYLEG6#%$BOXG" 1 2 0 1 10 0 2 1 6 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 868 "pde := diff(u(x,y),x$2)+diff(u(x,y),y$2)=0;\nu(x,y)=x^2-y^2;\nsimplify(ev al(subs(%,pde)));\nu(x,y)=y^3-3*y*x^2;\nsimplify(eval(subs(%,pde)));\n u(x,y)=y^4-6*y^2*x^2+x^4;\nsimplify(eval(subs(%,pde)));\nu(x,y)=y^5-10 *y^3*x^2+5*y*x^4;\nsimplify(eval(subs(%,pde)));\nu(x,y)=y^6-15*y^4*x^2 +15*y^2*x^4-x^6;\nsimplify(eval(subs(%,pde)));\nu(x,y)=exp(-y)*cos(x); \nsimplify(eval(subs(%,pde)));\nu(x,y)=sin(y)*cosh(x);\nsimplify(eval( subs(%,pde)));\nu(x,y)=ln(x^2+y^2);\nsimplify(eval(subs(%,pde)));\nu(x ,y)=arctan(y/x);\nsimplify(eval(subs(%,pde)));\nu(x,y)=2*cos(y)*sin(y) /(cos(y)^2+sinh(x)^2);\nsimplify(eval(subs(%,pde)));\nu(x,y)=2*cosh(x) *sin(y)/(cosh(x)^2-cos(y)^2);\nsimplify(eval(subs(%,pde)));\nu(x,y)=ex p(y^2-x^2)*cos(2*x*y);\nsimplify(eval(subs(%,pde)));\nu(x,y)=2*cos(y^2 )*cos(x^2)*cosh(y*x)^2-cos(y^2)*cos(x^2)+2*sin(y^2)*sin(x^2)*cosh(y*x) ^2-sin(y^2)*sin(x^2);\nsimplify(eval(subs(%,pde)));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$pdeG/,&-%%diffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F-\"\"# \"\"\"-F(6$F*-F06$F.F2F3\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\" uG6$%\"xG%\"yG,&*$)F'\"\"#\"\"\"F-*$)F(F,F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$% \"xG%\"yG,&*$)F(\"\"$\"\"\"F-*(F,F-F(F-)F'\"\"#F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"u G6$%\"xG%\"yG,(*$)F(\"\"%\"\"\"F-*(\"\"'F-)F(\"\"#F-)F'F1F-!\"\"*$)F'F ,F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG,(*$)F(\"\"&\"\"\"F-*(\"#5F-)F(\"\"$ F-)F'\"\"#F-!\"\"*(F,F-F(F-)F'\"\"%F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG, **$)F(\"\"'\"\"\"F-*(\"#:F-)F(\"\"%F-)F'\"\"#F-!\"\"*(F/F-)F(F3F-)F'F1 F-F-*$)F'F,F-F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG*&-%$expG6#,$F(!\"\"\"\" \"-%$cosG6#F'F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG*&-%$sinG6#F(\"\"\"-%%cos hG6#F'F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG-%#lnG6#,&*$)F'\"\"#\"\"\"F0*$)F( F/F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG-%'arctanG6#*&F(\"\"\"F'!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG,$**\"\"#\"\"\"-%$cosG6#F(F,-%$sinGF/F,,&*$)F -F+F,F,*$)-%%sinhG6#F'F+F,F,!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG,$**\" \"#\"\"\"-%%coshG6#F'F,-%$sinG6#F(F,,&*$)-%$cosGF2F+F,!\"\"*$)F-F+F,F, F8F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG*&-%$expG6#,&*$)F(\"\"#\"\"\"F1*$)F' F0F1!\"\"F1-%$cosG6#,$*(F0F1F(F1F'F1F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG, ***\"\"#\"\"\"-%$cosG6#*$)F(F+F,F,-F.6#*$)F'F+F,F,)-%%coshG6#*&F(F,F'F ,F+F,F,*&F-F,F2F,!\"\"**F+F,-%$sinGF/F,-F?F3F,F6F,F,*&F>F,F@F,F<" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 8 "The PDE " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Dif f(u(x,y),x)=0" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yGF*\"\"!" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 25 "has the general solution " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y)=f(x)" "6#/-%\"uG6$% \"xG%\"yG-%\"fG6#F'" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "whe re " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 29 " is an arbi trary function of " }{TEXT 265 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Similarly, the PDE " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),y) = 0; " "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yGF+\"\"!" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 25 "has the general solution " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y) = g(y);" "6#/-%\"uG6$%\"xG%\"yG- %\"gG6#F(" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "g(y);" "6#-%\"gG6#%\"yG" }{TEXT -1 29 " is an arbitrary function of " }{TEXT 266 1 "y" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 8 "The PDE " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),`$`(x,2))" "6#- %%DiffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F)\"\"#" }{XPPEDIT 18 0 "``+u(x,y) \+ = 0;" "6#/,&%!G\"\"\"-%\"uG6$%\"xG%\"yGF&\"\"!" }{TEXT -1 13 " ------- (i)," }}{PARA 0 "" 0 "" {TEXT -1 18 "or, more succintly" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[xx]+u=0" "6#/,&&%\"uG6#%#xxG \"\"\"F&F)\"\"!" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 64 "may \+ be compared with the ordinary differential equation ( ODE ) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x),`$`(x,2));" "6#-% %DiffG6$-%\"uG6#%\"xG-%\"$G6$F)\"\"#" }{XPPEDIT 18 0 "``+u(x)=0" "6#/, &%!G\"\"\"-%\"uG6#%\"xGF&\"\"!" }{TEXT -1 14 " ------- (ii)." }}{PARA 0 "" 0 "" {TEXT -1 38 "The ODE (ii) has the general solution " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x)=A*sin*x+B*cos*x " "6#/-%\"uG6#%\"xG,&*(%\"AG\"\"\"%$sinGF+F'F+F+*(%\"BGF+%$cosGF+F'F+F +" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 268 1 "A" }{TEXT -1 5 " and " }{TEXT 271 1 "B" }{TEXT -1 26 " are arbitrary \+ constants. " }}{PARA 0 "" 0 "" {TEXT -1 71 "This solution can be obtai ned with reference to the auxiliary equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m^2+1=0" "6#/,&*$%\"mG\"\"#\"\"\"F(F(\" \"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 16 "which has roots \+ " }{XPPEDIT 18 0 "m = 0;" "6#/%\"mG\"\"!" }{TEXT -1 1 " " }{TEXT 270 1 "+" }{TEXT -1 1 " " }{TEXT 269 1 "i" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 83 "The solution of the ODE (ii) is then obtained with r eference to the standard form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "u(x)=A*exp(p*x)*cos*q*x+B*exp(p*x)*sin*q*x" "6#/-%\"uG6 #%\"xG,&*,%\"AG\"\"\"-%$expG6#*&%\"pGF+F'F+F+%$cosGF+%\"qGF+F'F+F+*,% \"BGF+-F-6#*&F0F+F'F+F+%$sinGF+F2F+F'F+F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 42 "for the case where the auxiliary equation " } {XPPEDIT 18 0 "a*m^2+b*m+c=0" "6#/,(*&%\"aG\"\"\"*$%\"mG\"\"#F'F'*&%\" bGF'F)F'F'%\"cGF'\"\"!" }{TEXT -1 36 " associated with a second order \+ ODE " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 272 1 "a" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x),`$`(x,2))" "6#-%%DiffG6$-%\"uG6#%\"xG-% \"$G6$F)\"\"#" }{XPPEDIT 18 0 "`` + b" "6#,&%!G\"\"\"%\"bGF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x),x)" "6#-%%DiffG6$-%\"uG6#%\"xGF)" } {XPPEDIT 18 0 "``+c*u(x) = 0;" "6#/,&%!G\"\"\"*&%\"cGF&-%\"uG6#%\"xGF& F&\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 18 "has complex ro ots " }{XPPEDIT 18 0 "m = p" "6#/%\"mG%\"pG" }{TEXT -1 1 " " }{TEXT 273 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "q*i" "6#*&%\"qG\"\"\"%\"iGF% " }{TEXT -1 7 ", with " }{TEXT 274 1 "p" }{TEXT -1 5 " and " }{TEXT 275 1 "q" }{TEXT -1 7 " real. " }}{PARA 0 "" 0 "" {TEXT -1 84 "Since t he PDE (i) only involves a (second order) partial derivative with resp ect to " }{TEXT 276 1 "x" }{TEXT -1 82 ", a parallel argument to that \+ just given shows that (i) has the general solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y)=f(y)*cos*x+g(y)*sin*x" "6#/ -%\"uG6$%\"xG%\"yG,&*(-%\"fG6#F(\"\"\"%$cosGF.F'F.F.*(-%\"gG6#F(F.%$si nGF.F'F.F." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "f(y)" "6#-%\"fG6#%\"yG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(y)" "6#-%\"gG6#%\"yG" }{TEXT -1 28 " are arbitrary functions o f " }{TEXT 277 1 "y" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 81 "Th e PDE (i) is second order and linear homogeneous with constant coeffic ients in " }{TEXT 281 1 "u" }{TEXT -1 18 " as a function of " }{TEXT 282 1 "x" }{TEXT -1 8 ", where " }{TEXT 283 1 "y" }{TEXT -1 57 " is th ought of as being a temporarily constant parameter." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "u := 'u': \npde := Diff(u(x,y),x$2)+u(x,y)=0;\npdsolve(pde);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$pdeG/,&-%%DiffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F-\"\"# \"\"\"F*F3\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG ,&*&-%$_F1G6#F(\"\"\"-%$sinG6#F'F.F.*&-%$_F2GF-F.-%$cosGF1F.F." }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 8 "Th e PDE " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y) ,`$`(x,2))" "6#-%%DiffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F)\"\"#" }{XPPEDIT 18 0 "``+``;" "6#,&%!G\"\"\"F$F%" }{XPPEDIT 18 0 "Diff(u(x,y),x)" "6#- %%DiffG6$-%\"uG6$%\"xG%\"yGF)" }{XPPEDIT 18 0 "`` -2*u(x,y)=0" "6#/,&% !G\"\"\"*&\"\"#F&-%\"uG6$%\"xG%\"yGF&!\"\"\"\"!" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 18 "or, more succintly" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[xx]+u[x]-2*u = 0;" "6#/,(&%\"uG6#%#xx G\"\"\"&F&6#%\"xGF)*&\"\"#F)F&F)!\"\"\"\"!" }{TEXT -1 1 "," }}{PARA 257 "" 0 "" {TEXT -1 69 "is second order and linear homogeneous with c onstant coefficients in " }{TEXT 278 1 "u" }{TEXT -1 18 " as a functio n of " }{TEXT 279 1 "x" }{TEXT -1 7 " where " }{TEXT 280 1 "y" }{TEXT -1 58 " is thought of as being a temporarily constant parameter. " }} {PARA 0 "" 0 "" {TEXT -1 71 "This solution can be obtained with refere nce to the auxiliary equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "m^2+m-2 = 0;" "6#/,(*$%\"mG\"\"#\"\"\"F&F(F'!\"\"\"\"! " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 30 "The quadratic factor s to give " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(m+2)*( m-1)=0" "6#/*&,&%\"mG\"\"\"\"\"#F'F',&F&F'F'!\"\"F'\"\"!" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 17 "and so has roots " }{XPPEDIT 18 0 " m = -2;" "6#/%\"mG,$\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "m= 1" "6#/%\"mG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 34 "Th e PDE has the general solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "u(x,y) = f(y)*exp(-2*x)+g(y)*exp(x);" "6#/-%\"uG6$%\"xG %\"yG,&*&-%\"fG6#F(\"\"\"-%$expG6#,$*&\"\"#F.F'F.!\"\"F.F.*&-%\"gG6#F( F.-F06#F'F.F." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "f(y)" "6#-%\"fG6#%\"yG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(y)" "6#-%\"gG6#%\"yG" }{TEXT -1 28 " are arbitrary functions o f " }{TEXT 267 1 "y" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "Fo r example, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y) = exp(-2*x)*sin*y-exp(x)*y^2;" "6#/-%\"uG6$%\"xG%\"yG,&*(-%$expG6#,$* &\"\"#\"\"\"F'F1!\"\"F1%$sinGF1F(F1F1*&-F,6#F'F1*$F(F0F1F2" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 37 "is a particular solution of the \+ PDE. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "u := (x,y) -> exp(-2*x)*sin(2*y)-exp(x)*y^2;\nplot3d (u(x,y),x=-2.3..2.3,y=-3..3,grid=[40,40],\n color=COLOR(RGB,.8, .6,1),axes=boxed,\n lightmodel=light4,labels=[`x`,`y`,`u`]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6$%\"xG%\"yG6\"6$%)operatorG%& arrowGF),&*&-%$sinG6#,$*&\"\"#\"\"\"9%F5F5F5-%$expG6#,$*&F4F59$F5!\"\" F5F5*&)F6F4F5-F86#F " 0 "" {MPLTEXT 1 0 74 "u := 'u':\npde := Diff(u(x,y),x$2)+Diff(u(x,y),x)-2*u(x,y)=0;\npds olve(pde);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/,(-%%DiffG6$-%\" uG6$%\"xG%\"yG-%\"$G6$F-\"\"#\"\"\"-F(6$F*F-F3*&F2F3F*F3!\"\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG,&*&-%$_F1G6#F(\"\" \"-%$expG6#,$*&\"\"#F.F'F.!\"\"F.F.*&-%$_F2GF-F.-F06#F'F.F." }}}{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 -1 296 "From the e xamples considered so far it appears that the general solution of a pa rtial differential equation usually contains arbitrary functions. We c an consider the converse problem to that of solving a given PDE, namel y, that of the construction of a PDE with a given solution. For exampl e, if " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y) = x* f(y);" "6#/-%\"uG6$%\"xG%\"yG*&F'\"\"\"-%\"fG6#F(F*" }{TEXT -1 13 " -- ----- (i)," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "f(y) ;" "6#-%\"fG6#%\"yG" }{TEXT -1 29 " is an arbitrary function of " } {TEXT 284 1 "y" }{TEXT -1 7 ", then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),x) = f(y);" "6#/-%%DiffG6$-%\"uG6$%\"xG %\"yGF*-%\"fG6#F+" }{TEXT -1 15 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 40 "We can eliminate the arbitrary function " }{XPPEDIT 18 0 "f(y)" "6#-%\"fG6#%\"yG" }{TEXT -1 33 " from the equations (i) and (ii )." }}{PARA 0 "" 0 "" {TEXT -1 29 "Multiplying equation (ii) by " } {TEXT 285 1 "x" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 286 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),x) = \+ x*f(y)" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yGF**&F*\"\"\"-%\"fG6#F+F-" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 50 "and then making use of \+ equation (i) to substitute " }{XPPEDIT 18 0 "u(x,y)" "6#-%\"uG6$%\"xG% \"yG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "x*f(y)" "6#*&%\"xG\"\"\"-%\" fG6#%\"yGF%" }{TEXT -1 35 " on the right side gives the PDE: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 287 1 "x" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Diff(u(x,y),x)=u(x,y)" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yG F*-F(6$F*F+" }{TEXT -1 16 " ------- (iii), " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x*u[x]=u " "6#/*&%\"xG\"\"\"&%\"uG6#F%F&F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "Any function " }{XPPEDIT 18 0 "u(x,y)" "6#-%\"uG6$%\"xG% \"yG" }{TEXT -1 84 " of the form given by (i) is a solution of the par tial differential equation (iii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "u := 'u':\npde := x*diff(u(x ,y),x)=u(x,y);\npdsolve(pde);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pd eG/*&%\"xG\"\"\"-%%diffG6$-%\"uG6$F'%\"yGF'F(F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG*&-%$_F1G6#F(\"\"\"F'F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose tha t " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y)=f(x+y)+g (x-y)" "6#/-%\"uG6$%\"xG%\"yG,&-%\"fG6#,&F'\"\"\"F(F.F.-%\"gG6#,&F'F.F (!\"\"F." }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(t)" "6#-%\"gG6#%\"tG" }{TEXT -1 46 " are arbitrary functions o f a single variable " }{TEXT 288 1 "t" }{TEXT -1 6 ", say." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Diff(u(x,y),x)=``" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yGF*%! G" }{TEXT -1 5 "f '( " }{XPPEDIT 18 0 "x+y" "6#,&%\"xG\"\"\"%\"yGF%" } {TEXT -1 2 " )" }{XPPEDIT 18 0 "`` + ``" "6#,&%!G\"\"\"F$F%" }{TEXT -1 5 "g '( " }{XPPEDIT 18 0 "x-y" "6#,&%\"xG\"\"\"%\"yG!\"\"" }{TEXT -1 3 " ) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x ,y),x$2)=``" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F*\"\"#%!G" } {TEXT -1 5 "f \"( " }{XPPEDIT 18 0 "x+y" "6#,&%\"xG\"\"\"%\"yGF%" } {TEXT -1 2 " )" }{XPPEDIT 18 0 "`` + ``" "6#,&%!G\"\"\"F$F%" }{TEXT -1 5 "g \"( " }{XPPEDIT 18 0 "x-y" "6#,&%\"xG\"\"\"%\"yG!\"\"" }{TEXT -1 3 " ) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x ,y),y)=``" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yGF+%!G" }{TEXT -1 5 "f '( " }{XPPEDIT 18 0 "x+y" "6#,&%\"xG\"\"\"%\"yGF%" }{TEXT -1 2 " )" } {XPPEDIT 18 0 "`` - ``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 5 "g '( " } {XPPEDIT 18 0 "x-y" "6#,&%\"xG\"\"\"%\"yG!\"\"" }{TEXT -1 3 " ) " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),y$2)=``" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F+\"\"#%!G" }{TEXT -1 5 "f \"( " }{XPPEDIT 18 0 "x+y" "6#,&%\"xG\"\"\"%\"yGF%" }{TEXT -1 2 " )" } {XPPEDIT 18 0 "`` + ``" "6#,&%!G\"\"\"F$F%" }{TEXT -1 5 "g \"( " } {XPPEDIT 18 0 "x-y" "6#,&%\"xG\"\"\"%\"yG!\"\"" }{TEXT -1 3 " ) " }} {PARA 0 "" 0 "" {TEXT -1 106 "Since the right hand sides of the second and last of these four equations are identical we obtain the PDE " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),`$`(x,2)) =Diff(u(x,y),`$`(y,2))" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F*\"\" #-F%6$-F(6$F*F+-F-6$F+F/" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[xx]=u[yy] " "6#/&%\"uG6#%#xxG&F%6#%#yyG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "For any choice of the fun ctions " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "g(t)" "6#-%\"gG6#%\"tG" }{TEXT -1 16 ", the function \+ " }{XPPEDIT 18 0 "u(x,y)=f(x+y)+g(x-y)" "6#/-%\"uG6$%\"xG%\"yG,&-%\"fG 6#,&F'\"\"\"F(F.F.-%\"gG6#,&F'F.F(!\"\"F." }{TEXT -1 29 " is a soluti on of this PDE. " }}{PARA 0 "" 0 "" {TEXT -1 20 "For example, taking \+ " }{XPPEDIT 18 0 "f(t)=g(t)" "6#/-%\"fG6#%\"tG-%\"gG6#F'" }{XPPEDIT 18 0 "``=y^2/2" "6#/%!G*&%\"yG\"\"#F'!\"\"" }{TEXT -1 25 " we obtain \+ the solution " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x, y) = 1/2" "6#/-%\"uG6$%\"xG%\"yG*&\"\"\"F*\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(x+y)^2 + 1/2" "6#,&*$,&%\"xG\"\"\"%\"yGF'\"\"#F'*&F'F 'F)!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(x-y)^2" "6#*$,&%\"xG\"\" \"%\"yG!\"\"\"\"#" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y)= 1/2 " "6#/-%\"uG6$%\"xG%\"yG*&\"\"\"F*\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "``(x^2+2*x*y+y^2) +1/2" "6#,&-%!G6#,(*$%\"xG\"\"#\"\"\" *(F*F+F)F+%\"yGF+F+*$F-F*F+F+*&F+F+F*!\"\"F+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "``(x^2-2*x*y+y^2)" "6#-%!G6#,(*$%\"xG\"\"#\"\"\"*(F)F*F (F*%\"yGF*!\"\"*$F,F)F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y)=x^2+ y^2" "6#/-%\"uG6$%\"xG%\"yG,&*$F'\"\"#\"\"\"*$F(F+F," }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Lettin g " }{XPPEDIT 18 0 "f(t)=g(t)" "6#/-%\"fG6#%\"tG-%\"gG6#F'" }{XPPEDIT 18 0 "`` = sin*t/2;" "6#/%!G*(%$sinG\"\"\"%\"tGF'\"\"#!\"\"" }{TEXT -1 24 " we obtain the solution " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y)=1/2" "6#/-%\"u G6$%\"xG%\"yG*&\"\"\"F*\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin (x+y) + 1/2" "6#,&-%$sinG6#,&%\"xG\"\"\"%\"yGF)F)*&F)F)\"\"#!\"\"F)" } {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(x-y)" "6#-%$sinG6#,&%\"xG\"\"\"%\"y G!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y)= 1/2" "6#/-%\" uG6$%\"xG%\"yG*&\"\"\"F*\"\"#!\"\"" }{XPPEDIT 18 0 "``( sin*x*cos*y+co s*x*sin*y + sin*x*cos*y-cos*x*sin*y)" "6#-%!G6#,***%$sinG\"\"\"%\"xGF) %$cosGF)%\"yGF)F)**F+F)F*F)F(F)F,F)F)**F(F)F*F)F+F)F,F)F)**F+F)F*F)F(F )F,F)!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "or, " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y)=sin*x*cos*y" " 6#/-%\"uG6$%\"xG%\"yG**%$sinG\"\"\"F'F+%$cosGF+F(F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "u := (x,y) -> sin(x)*cos(y);\nplot3d(u(x,y),x=-2*Pi..2*Pi,y=-2* Pi..2*Pi,grid=[50,50],\n color=COLOR(RGB,.8,.6,1),axes=boxed,\n lightmodel=light4,labels=[`x`,`y`,`u`],orientation=[43,22]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6$%\"xG%\"yG6\"6$%)operatorG%& arrowGF)*&-%$sinG6#9$\"\"\"-%$cosG6#9%F2F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT3D 712 396 396 {PLOTDATA 3 "6'-%%GRIDG6&;$!0ezrI&=$G'!#9$\"0ezr I&=$G'F)F&X,%)anythingG6\"6\"[gl'!%\"!!#_gt\"S\"S3CFD1A62633145C03CFC2 6B7EEB733623CF95BA8C26498FB3CF4E7FA8E9CEBEA3CEE2C729E85E8993CE08FAE03A C14F73CADD97D82CFF170BCD9E77746285C47BCEAEC2D11FB3780BCF390E667655D73B CF86414174CFCEFBCFB9ED47EF2DA6CBCFD0B13A36CA316BCFC90FE32849D7CBCFA389 0780A9536BCF62910A606F166BCF0A67D2EEC7DB1BCE41A346C3AD815BCC65B447FF23 E2D3CD294527D604B603CE78F953C37ADFD3CF2253D17FC09623CF752D6E89FA09E3CF AF9E2D62BC7C83CFCDD377E7BAAC83CFCDD377E7BAAC23CFAF9E2D62BC7B83CF752D6E 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84F4C1DBCC974237098B769BCC0866BC15CE111BCAA0FF62E2B40263C9F5C59B7F3955 C3CBC32DF34F52EB73CC75B3BE0D968F83CCF15EDB3911E843CD2640BE2C83FC93CD40 92C8AAF54583CD45ECD064D1E093CD35F53827C36273CD11B75ED92F3A23CCB72444A9 C0F873CC2E205ADA04A783CB22B2A6F9A9DDABC84EF9329C8AB88BCB73B3CF24ADF00B CC529C2688E1307BCCD5385CC9309D4BCD1C91BBF987F8EBCD3BEA2A8FC06BEBCD4698 98CC51701-%&COLORG6&%$RGBG$\"\")!\"\"$\"\"'F6$\"\"\"\"\"!-%+LIGHTMODEL G6#%(LIGHT_4G-%*AXESSTYLEG6#%$BOXG-%+AXESLABELSG6%%\"xG%\"yG%\"uG-%+PR OJECTIONG6%$\"#VF;$\"#AF;F:" 1 2 0 1 10 0 2 1 6 2 2 1.000000 22.000000 43.000000 1 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "u := 'u':\npde := diff(u(x,y ),x$2)=diff(u(x,y),y$2);\npdsolve(pde);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/-%%diffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F,\"\"#-F'6$F)-F/6$F -F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG,&-%$_F1G6#, &F(\"\"\"F'F.F.-%$_F2G6#,&F(F.F'!\"\"F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "f := x -> x^4/2;\ng : = x -> x^4/2;\nf(x+y)+g(x-y);\nexpand(%);\nsimplify(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&#\" \"\"\"\"#F/*$)9$\"\"%F/F/F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&#\"\"\"\"\"#F/*$)9$\"\"%F/ F/F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#!\"\",&%\"yG\" \"\"%\"xGF)\"\"%F)*&F%F&,&F*F)F(F&F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"yG\"\"%\"\"\"F(*(\"\"'F()F&\"\"#F()%\"xGF,F(F(*$)F.F'F(F (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"yG\"\"%\"\"\"F(*(\"\"'F() F&\"\"#F()%\"xGF,F(F(*$)F.F'F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Example 6 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 17 "C onsider the PDE " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "D iff(u(x,y),x,y) = 0;" "6#/-%%DiffG6%-%\"uG6$%\"xG%\"yGF*F+\"\"!" } {TEXT -1 13 " ------- (i)," }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[xy]=0" "6#/&%\"uG6# %#xyG\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "in which we assume that " }{XPPEDIT 18 0 "u(x,y )" "6#-%\"uG6$%\"xG%\"yG" }{TEXT -1 46 " has continuous partial deriva tives, so that " }{XPPEDIT 18 0 "Diff(u(x,y),y,x)=Diff(u(x,y),x,y)" " 6#/-%%DiffG6%-%\"uG6$%\"xG%\"yGF+F*-F%6%-F(6$F*F+F*F+" }{XPPEDIT 18 0 "``=0" "6#/%!G\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Le t " }{XPPEDIT 18 0 "p=u[x]" "6#/%\"pG&%\"uG6#%\"xG" }{TEXT -1 12 ", t hat is, " }{XPPEDIT 18 0 "p(x,y)=Diff(u(x,y),x)" "6#/-%\"pG6$%\"xG%\" yG-%%DiffG6$-%\"uG6$F'F(F'" }{TEXT -1 20 ". Then (i) becomes: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(p(x,y),y)=0" "6# /-%%DiffG6$-%\"pG6$%\"xG%\"yGF+\"\"!" }{TEXT -1 15 " ------- (ii). " } }{PARA 0 "" 0 "" {TEXT -1 33 "Integrating (ii) with respect to " } {TEXT 321 1 "y" }{TEXT -1 11 ", treating " }{TEXT 322 1 "x" }{TEXT -1 31 " as a constant parameter gives:" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "p(x,y) = h(x);" "6#/-%\"pG6$%\"xG%\"yG-%\"hG6#F'" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "or " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),x) = h(x);" "6#/-%%DiffG6 $-%\"uG6$%\"xG%\"yGF*-%\"hG6#F*" }{TEXT -1 15 " ------- (ii), " }} {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "h(x);" "6#-%\"hG6# %\"xG" }{TEXT -1 40 " is an arbitrary continuous function of " }{TEXT 324 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " } {XPPEDIT 18 0 "h(x);" "6#-%\"hG6#%\"xG" }{TEXT -1 41 " is continuous, \+ we can obtain a function " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" } {TEXT -1 15 " such that f '(" }{TEXT 323 1 "x" }{TEXT -1 1 ")" } {XPPEDIT 18 0 "`` = h(x);" "6#/%!G-%\"hG6#%\"xG" }{TEXT -1 40 ". Then \+ integrating (ii) with respect to " }{TEXT 325 1 "x" }{TEXT -1 11 ", tr eating " }{TEXT 326 1 "y" }{TEXT -1 22 " as a constant gives: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y)=f(x)+g(y)" "6# /-%\"uG6$%\"xG%\"yG,&-%\"fG6#F'\"\"\"-%\"gG6#F(F-" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "g(y);" "6#-%\"gG6# %\"yG" }{TEXT -1 29 " is an arbitrary function of " }{TEXT 327 1 "y" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 74 "Note that differentiatin g this equation partially with respect to y gives " }{XPPEDIT 18 0 "Di ff(u(x,y),y) = ``;" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yGF+%!G" }{TEXT -1 4 "g '(" }{TEXT 329 1 "y" }{TEXT -1 56 "), which, by our hypothesis, i s a continous function of " }{TEXT 328 1 "y" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 161 "In future we shall consider suitable continuit y assumptions to be implied where integration of arbitrary functions a nd/or mixed partial derivatives are involved." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 15 "As an example, " } {XPPEDIT 18 0 "u(x,y) = sin*x + y^2" "6#/-%\"uG6$%\"xG%\"yG,&*&%$sinG \"\"\"F'F,F,*$F(\"\"#F," }{TEXT -1 23 " is a solution of (i). 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19EF401A75860BFBAD094018179B410038664015DB59F312BC044013C0C2223337E540 11C7D3CE61AC07400FE11DEF3C30D7400C75E73BD0FA2440094E038281B3F440066972 C34E5E484003C834FE36F92140016A4A333B847D3FFE9F64C4B800BB3FFAF0DB1730D9 843FF7C8F75DE193543FF527B998CA2E2C3FF30D21C7EAAA0D3FF1792FEB4306F53FF0 6BE402D344E53FEFCA7C1D36C7B93FEFCA7C1D36C7B93FF06BE402D344E43FF1792FEB 4306F43FF30D21C7EAAA0B3FF527B998CA2E2A3FF7C8F75DE193513FFAF0DB1730D980 3FFE9F64C4B800B840016A4A333B847B4003C834FE36F91E40066972C34E5E4640094E 038281B3F1400C75E73BD0FA20400FE11DEF3C30D34011C7D3CE61AC054013C0C22233 37E34015DB59F312BC024018179B41003864401A75860BFBAD07401CF51A540519ED-% &COLORG6&%$RGBG$F+F/$\"\"'F/$\"\"\"F)-%*AXESSTYLEG6#%$BOXG-%+LIGHTMODE LG6#%(LIGHT_4G-%+AXESLABELSG6%%\"xG%\"yG%\"uG" 1 2 0 1 10 0 2 1 6 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "u := 'u':\npde := \+ diff(u(x,y),x,y)=0;\npdsolve(pde);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%$pdeG/-%%diffG6%-%\"uG6$%\"xG%\"yGF,F-\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG,&-%$_F2G6#F'\"\"\"-%$_F1G6#F(F-" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 7 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 43 "C onsider the partial differential equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),x,y)" "6#-%%DiffG6%-%\"uG6$%\"xG% \"yGF)F*" }{XPPEDIT 18 0 "``+``" "6#,&%!G\"\"\"F$F%" }{XPPEDIT 18 0 "D iff(u(x,y),x)=0" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yGF*\"\"!" }{TEXT -1 12 " ------- (i)" }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[xy]+u[x]=0" "6#/,&&%\"uG6#%#xyG\" \"\"&F&6#%\"xGF)\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 " Let " }{XPPEDIT 18 0 "p=u[x]" "6#/%\"pG&%\"uG6#%\"xG" }{TEXT -1 12 ", that is, " }{XPPEDIT 18 0 "p(x,y)=Diff(u(x,y),x)" "6#/-%\"pG6$%\"xG% \"yG-%%DiffG6$-%\"uG6$F'F(F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "Then equation (i) is " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Diff(p(x,y),y)" "6#-%%DiffG6$-%\"pG6$%\"xG%\"yGF*" } {XPPEDIT 18 0 "``+p(x,y)=0" "6#/,&%!G\"\"\"-%\"pG6$%\"xG%\"yGF&\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(p(x,y),y)=-p(x,y)" "6#/-%% DiffG6$-%\"pG6$%\"xG%\"yGF+,$-F(6$F*F+!\"\"" }{TEXT -1 14 " ------- (i i)." }}{PARA 0 "" 0 "" {TEXT -1 12 "Considering " }{TEXT 289 1 "y" } {TEXT -1 122 " to be a parameter, which we temporarily take to be cons tant, makes equation (ii) a first order differential equation, say" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dp/dy = -p;" "6#/*&%# dpG\"\"\"%#dyG!\"\",$%\"pGF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 92 "This differential equation has separable variables, and h as a general solution of the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "p = ``;" "6#/%\"pG%!G" }{TEXT -1 12 "'constant' " } {XPPEDIT 18 0 "`.`*exp(-y)" "6#*&%\".G\"\"\"-%$expG6#,$%\"yG!\"\"F%" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 72 "In our context the cons tant should be replaced by an arbitrary function " }{XPPEDIT 18 0 "h(x );" "6#-%\"hG6#%\"xG" }{TEXT -1 4 " of " }{TEXT 290 1 "x" }{TEXT -1 24 ", to give the solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "p(x,y) = h(x)*exp(-y);" "6#/-%\"pG6$%\"xG%\"yG*&-%\"hG6 #F'\"\"\"-%$expG6#,$F(!\"\"F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "for (ii)." }}{PARA 0 "" 0 "" {TEXT -1 20 "Now, recalling t hat " }{XPPEDIT 18 0 "Diff(u(x,y),x)=p(x,y)" "6#/-%%DiffG6$-%\"uG6$%\" xG%\"yGF*-%\"pG6$F*F+" }{TEXT -1 14 ", we obtain: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),x) = h(x)*exp(-y);" "6#/ -%%DiffG6$-%\"uG6$%\"xG%\"yGF**&-%\"hG6#F*\"\"\"-%$expG6#,$F+!\"\"F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Treating " }{TEXT 291 1 "y" }{TEXT -1 47 " as a constant a nd integrating with respect to " }{TEXT 292 1 "x" }{TEXT -1 13 " we se e that " }{XPPEDIT 18 0 "u(x,y)" "6#-%\"uG6$%\"xG%\"yG" }{TEXT -1 15 " has the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u( x,y)=f(x)*exp(-y)+g(y)" "6#/-%\"uG6$%\"xG%\"yG,&*&-%\"fG6#F'\"\"\"-%$e xpG6#,$F(!\"\"F.F.-%\"gG6#F(F." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 10 "where f '(" }{TEXT 294 1 "x" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "``=h(x)" "6#/%!G-%\"hG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(y)" "6#-%\"gG6#%\"yG" }{TEXT -1 30 " is an arbitrary function of \+ " }{TEXT 293 1 "y" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "Fo r example, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y) = exp(-y)*sin*x+8*y^2;" "6#/-%\"uG6$%\"xG%\"yG,&*(-%$expG6#,$F(!\"\" \"\"\"%$sinGF0F'F0F0*&\"\")F0*$F(\"\"#F0F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "is a solution of (i). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "u := (x,y) -> sin (x)*exp(-y)+8*y^2;\nplot3d(u(x,y),x=-7..7,y=-4..4,grid=[40,40],\n \+ color=COLOR(RGB,.8,.6,1),axes=boxed,\n lightmodel=light4,label s=[`x`,`y`,`u`],orientation=[33,33]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&*&-%$sinG6#9$\"\"\" -%$expG6#,$9%!\"\"F3F3*&\"\")F3)F8\"\"#F3F3F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT3D 592 351 351 {PLOTDATA 3 "6'-%%GRIDG6&;$!\"(\"\"!$\"\"(F);$! \"%F)$\"\"%F)X,%)anythingG6\"6\"[gl'!%\"!!#[_q\"I\"I4057084DC6D62FB040 557F654698F58C4053D29DAA76D6B14052109BAEF4B6B04050454CD5BFDA4A404CF4D0 AA3EC7A940496FB03BDD5EC04046081CCECB020D4042C896BACF859F403F73578A5D61 AC4039C4A72EC6070E403490761FE171BA402FC0066DD2A820402775ACB4572C2E4020 4E23CA0E25754014A6D3DBEF28634006368BB4ACF1B23FF01B8100A37294BFC171FA72 4D7710BFE499F08C667E63BFE047FEFB713E5D3FD19011FA5F48203FFB5E18929D0974 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402C6F933975A32B40245BC94EE2E98A401B413E28B6DE6540107C304554C06B4000D0 DD9F5979393FE82FF46CD0C0EA3FB538D8977F01493FB5481E297628C23FE835C32634 84664000D35A813CE51540107E00C24B69A8401B43B44EAAE4F440245D648435B84540 2C719FEE2971B54032EF4662CF73FA40385215AC91126D403E613DF457023940428E5F AA17F3D24046424CF0AAA461404A4C66D6887E10404EACAD62BC05554051B1904D80BA 40405437E042025D644056E94690C9FDFE4059C5C33B645958405CCD56431498B4405F FFFFA8E1F69F406263F09C9781C14060591EDE244C0E405CF1C8D5D73AA040597EB3FF D986A7405652A3583F84FF40536868E40393CD4050BBCC91BBDA0E404C92BD3CD0CA7F 40481CA4DF0187F2404410C0A1632CCA40406B599C66E9DA403A52D28536F005403490 F15803ED36402F1A17C8A0A164402687B9F43C7618401ECAEAB26B68BF40135DE61AE4 DE21400582A8B70763833FF3BDCD427F534A3FDE3923A4FA29093FD99DF08AE319513F F03CF43D416BE940028246C2F5387A40112D816B7518F1401BD2A8A0CB58054024979D 2719F1D5402CA10C654D3F3740330296D9CE7608403861D127186275403E6E0E822CE2 9440429397BD8AD5054046468D3B34D9BE404A4FDD4C3FAAB2404EAF7F71FA17F94051 B2B660816CFA405438CFCB3E410B4056EA09AD9048424059C66228CF8C6F405CCDD7B7 1941F8406000348D6C9E5140647BD91C94E82040620DA40ABBEF93405FB8E9D2ED4E49 405BC1F2B5053A1D40582A753A125DE24054E8BA2BCD97404051F4D7995B8A26404E90 B69708E4F64049BC0A2CC8CC714045631C2EB3035940417EF4F34D09E0403C13CF3825 49A94035FEA978CDF6124030B6F0C24630F240286D05725E3EBE4020F0C07C11827040 15E1DDA91CD1EA40099BBC8DA6887F3FFA6AD6C8FC6D7E3FE9FCBCE08E36C33FE5AACB 4F98F6FB3FF3D892FC1AECF04003FA70C4A9BD5A4011C6B4D8FFBF0C401C4F7288AD95 014024CA6FCECCDD1A402CCA7221D68155403313730196AB5640386F8CD8ABAFD3403E 793E3115932C40429826096B8D5D40464A433ACDD4DC404A52E31CBFE622404EB1F5C0 124F394051B3B7150C987E405439A0E42F9BAF4056EAB3FF2ED1AA4059C6ECE4330757 405CCE48B7E7D64540600062934D1992-%&COLORG6&%$RGBG$\"\")!\"\"$\"\"'F;$ \"\"\"F)-%+LIGHTMODELG6#%(LIGHT_4G-%*AXESSTYLEG6#%$BOXG-%+AXESLABELSG6 %%\"xG%\"yG%\"uG-%+PROJECTIONG6%$\"#LF)FQF?" 1 2 0 1 10 0 2 1 6 2 2 1.000000 33.000000 33.000000 1 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "u := 'u':\npde := \+ diff(u(x,y),x,y)+diff(u(x,y),x)=0;\npdsolve(pde);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/,&-%%diffG6%-%\"uG6$%\"xG%\"yGF-F.\"\"\"-F(6$F* F-F/\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG,&-%$_ F1G6#F(\"\"\"*&-%$expG6#,$F(!\"\"F--%$_F2G6#F'F-F-" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 8 " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y)=f(x^2+y^2 )" "6#/-%\"uG6$%\"xG%\"yG-%\"fG6#,&*$F'\"\"#\"\"\"*$F(F.F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "f(t)" "6#- %\"fG6#%\"tG" }{TEXT -1 29 " is an arbitrary function of " }{TEXT 295 1 "t" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),x)=``" "6#/-%%D iffG6$-%\"uG6$%\"xG%\"yGF*%!G" }{TEXT -1 5 "f '( " }{XPPEDIT 18 0 "x^2 +y^2" "6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF&F'" }{TEXT -1 3 " ) " }{XPPEDIT 18 0 "2*x" "6#*&\"\"#\"\"\"%\"xGF%" }{TEXT -1 13 " ------- (i) " }} {PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Diff(u(x,y),y) = ``;" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yGF +%!G" }{TEXT -1 5 "f '( " }{XPPEDIT 18 0 "x^2+y^2" "6#,&*$%\"xG\"\"#\" \"\"*$%\"yGF&F'" }{TEXT -1 3 " ) " }{XPPEDIT 18 0 "2*y" "6#*&\"\"#\"\" \"%\"yGF%" }{TEXT -1 15 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 149 "Comparing the right sides of equations (i) and (ii) we notice tha t the arbitrary function can be eliminated by multiplying equations (i ) and (ii) by " }{TEXT 298 1 "y" }{TEXT -1 5 " and " }{TEXT 299 1 "x" }{TEXT -1 22 " respectively to give " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 296 1 "y" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),x)=`` " "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yGF*%!G" }{TEXT -1 5 "f '( " } {XPPEDIT 18 0 "x^2+y^2" "6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF&F'" }{TEXT -1 3 " ) " }{XPPEDIT 18 0 "2*x*y;" "6#*(\"\"#\"\"\"%\"xGF%%\"yGF%" } {TEXT -1 15 " ------- (iii) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 297 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),y) = ``;" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yGF+%!G" }{TEXT -1 5 "f '( " }{XPPEDIT 18 0 "x^2+y^2" "6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF&F'" }{TEXT -1 3 " ) " } {XPPEDIT 18 0 "2*x*y;" "6#*(\"\"#\"\"\"%\"xGF%%\"yGF%" }{TEXT -1 15 " \+ ------- (iv). " }}{PARA 0 "" 0 "" {TEXT -1 16 "It follows that " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 300 1 "x" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Diff(u(x,y),y)= y" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yGF+F+ " }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),x);" "6#-%%DiffG6$-%\"uG 6$%\"xG%\"yGF)" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 307 14 "______________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "For any choice of the function " }{XPPEDIT 18 0 "f(t)" "6#-%\"f G6#%\"tG" }{TEXT -1 16 ", the function " }{XPPEDIT 18 0 "u(x,y) = f(x ^2+y^2);" "6#/-%\"uG6$%\"xG%\"yG-%\"fG6#,&*$F'\"\"#\"\"\"*$F(F.F/" } {TEXT -1 29 " is a solution of this PDE. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 10 "Conversely" }{TEXT -1 27 ", we can show that the PDE " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 301 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),y)= y" "6#/-%%Di ffG6$-%\"uG6$%\"xG%\"yGF+F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x, y),x);" "6#-%%DiffG6$-%\"uG6$%\"xG%\"yGF)" }{TEXT -1 12 " ------- (v) " }}{PARA 0 "" 0 "" {TEXT -1 31 "only has solutions of the form " } {XPPEDIT 18 0 "u(x,y) = f(x^2+y^2)" "6#/-%\"uG6$%\"xG%\"yG-%\"fG6#,&*$ F'\"\"#\"\"\"*$F(F.F/" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 69 " We can make a change to polar coordinates by means of the relations: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x=r*co s*theta,``],[y=r*sin*theta,``])" "6#-%*PIECEWISEG6$7$/%\"xG*(%\"rG\"\" \"%$cosGF+%&thetaGF+%!G7$/%\"yG*(F*F+%$sinGF+F-F+F." }{TEXT -1 14 "--- ---- (vi). " }}{PARA 0 "" 0 "" {TEXT -1 65 "The partial differential e quation (v) can be written in the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 303 1 "y" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),x) " "6#-%%DiffG6$-%\"uG6$%\"xG%\"yGF)" }{XPPEDIT 18 0 "`` - x" "6#,&%!G \"\"\"%\"xG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),y)=0" "6 #/-%%DiffG6$-%\"uG6$%\"xG%\"yGF+\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "r*sin*theta" "6#*(%\"rG\"\"\"%$sinGF%%&thetaGF%" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u,x)" "6#-%%DiffG6$%\"uG%\"xG" } {XPPEDIT 18 0 "`` - r*cos*theta" "6#,&%!G\"\"\"*(%\"rGF%%$cosGF%%&thet aGF%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u,y)=0" "6#/-%%DiffG6$ %\"uG%\"yG\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u,x)" "6#-%%Dif fG6$%\"uG%\"xG" }{XPPEDIT 18 0 "``(-r*sin*theta)+``;" "6#,&-%!G6#,$*(% \"rG\"\"\"%$sinGF*%&thetaGF*!\"\"F*F%F*" }{XPPEDIT 18 0 "Diff(u,y)" "6 #-%%DiffG6$%\"uG%\"yG" }{XPPEDIT 18 0 "``(r*cos*theta)=0" "6#/-%!G6#*( %\"rG\"\"\"%$cosGF)%&thetaGF)\"\"!" }{TEXT -1 16 " ------- (vii). " }} {PARA 0 "" 0 "" {TEXT -1 24 "The relations (vi) give:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Diff(x,theta)=-r*sin*th eta,``],[Diff(y,theta)=r*cos*theta,``])" "6#-%*PIECEWISEG6$7$/-%%DiffG 6$%\"xG%&thetaG,$*(%\"rG\"\"\"%$sinGF0F,F0!\"\"%!G7$/-F)6$%\"yGF,*(F/F 0%$cosGF0F,F0F3" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 23 "so \+ that (vii) becomes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u,x)" "6#-%%DiffG6$%\"uG%\"xG" }{TEXT -1 1 " " }{TEXT 305 1 ". " }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(x,theta)" "6#-%%DiffG6$%\"xG%&t hetaG" }{XPPEDIT 18 0 "`` +`` " "6#,&%!G\"\"\"F$F%" }{XPPEDIT 18 0 "Di ff(u,y)" "6#-%%DiffG6$%\"uG%\"yG" }{TEXT -1 1 " " }{TEXT 306 1 "." } {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(y,theta)=0" "6#/-%%DiffG6$%\"yG%&t hetaG\"\"!" }{TEXT -1 18 " ------- (viii). " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 10 "chain rule" }{TEXT -1 20 " gives the relati on:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u,theta)= Diff(u,x)" "6#/-%%DiffG6$%\"uG%&thetaG-F%6$F'%\"xG" }{TEXT -1 1 " " } {TEXT 302 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(x,theta)" "6#-%%D iffG6$%\"xG%&thetaG" }{XPPEDIT 18 0 "`` + ``" "6#,&%!G\"\"\"F$F%" } {XPPEDIT 18 0 "Diff(u,y)" "6#-%%DiffG6$%\"uG%\"yG" }{TEXT -1 1 " " } {TEXT 304 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(y,theta)" "6#-%%D iffG6$%\"yG%&thetaG" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 41 "so that (viii) can be written simply as: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Diff(u,theta)=0" "6#/-%%DiffG6$%\"uG%&thetaG \"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u = h(r);" "6#/%\"uG- %\"hG6#%\"rG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "h(r);" "6#-%\"hG6#%\"rG" }{TEXT -1 29 " is an arbitrary function of " }{TEXT 308 1 "r" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "Since the relations (vi) imply that " }{XPPEDIT 18 0 "r^ 2=x^2+y^2" "6#/*$%\"rG\"\"#,&*$%\"xGF&\"\"\"*$%\"yGF&F*" }{TEXT -1 14 ", we see that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u( x,y)=h(sqrt(x^2+y^2))" "6#/-%\"uG6$%\"xG%\"yG-%\"hG6#-%%sqrtG6#,&*$F' \"\"#\"\"\"*$F(F1F2" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "th at is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y)=f(x ^2+y^2)" "6#/-%\"uG6$%\"xG%\"yG-%\"fG6#,&*$F'\"\"#\"\"\"*$F(F.F/" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 " f(t)=h(sqrt(t))" "6#/-%\"fG6#%\"tG-%\"hG6#-%%sqrtG6#F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "f(t)" "6 #-%\"fG6#%\"tG" }{TEXT -1 60 " can be any function of a non-negative r eal number argument " }{TEXT 310 1 "t" }{TEXT -1 25 ", since we can th en take " }{XPPEDIT 18 0 "h(r)" "6#-%\"hG6#%\"rG" }{TEXT -1 7 " to be \+ " }{XPPEDIT 18 0 "f(r^2)" "6#-%\"fG6#*$%\"rG\"\"#" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "The surf ace which constutes the graph of any solution has contours which are c ircles with centre at the origin. " }}{PARA 0 "" 0 "" {TEXT -1 17 "For example, the " }{XPPEDIT 18 0 "u(x,y)=exp(-x^2-y^2)" "6#/-%\"uG6$%\"x G%\"yG-%$expG6#,&*$F'\"\"#!\"\"*$F(F.F/" }{TEXT -1 19 " is a solution \+ of " }{TEXT 309 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),y)= y" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yGF+F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),y)" "6#-%%DiffG6$-%\"uG6$%\"xG%\"yGF*" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "u := (x,y) -> exp(-x^2-y^2);\nplot3d(u(x,y),x=-3..3, y=-3..3,grid=[40,40],\n color=COLOR(RGB,.8,.6,1),axes=boxed,\n \+ lightmodel=light4,labels=[`x`,`y`,`u`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)-%$expG6# ,&*$)9$\"\"#\"\"\"!\"\"*$)9%F4F5F6F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT3D 592 351 351 {PLOTDATA 3 "6&-%%GRIDG6&;$!\"$\"\"!$\"\"$F)F&X, %)anythingG6\"6\"[gl'!%\"!!#[_q\"I\"I3E505A628C699FA13E64196091F446D53 E778FA957CF97213E8A57B3C1CB357F3E9C1733B30732DC3EAC91DA8CECDA223EBBB6A 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1349FE7399EA03F64CAD32A430C423F67F709476458413F6A58406F8AAFF13F6B9F2EE 5FB97783F6B9F2EE5FB977E3F6A58406F8AAFF63F67F7094764584D3F64CAD32A430C5 13F61349FE7399EB23F5B2899D8FA31B43F54718FEE40A2BC3F4D5AD4777900203F441 9CBE8154FF83F3A415E73D8AA353F305AB9E12F47E83F236EA7E444416F3F1605961E6 C88B53F07CD79B5647D073EF889B86882BC603EE8206092AE51483ED6A0075DC1FB0D3 EC43C78B462DA323EB143428C3835563E9C1733B30733843E8A57B3C1CB35CE3EA0304 13852C7DA3EB2FA18D7EA6D933EC537990CA69A6D3ED6A0075DC1FAC93EE702D1069D2 B473EF652492CC90D9B3F04A6C4282F0ABF3F123916CA978FBF3F1EAC61E5BE5FE83F2 89F1B37A62F0D3F32D994700FA2BB3F3B8730F231884C3F432BE24DCFB24E3F4977F2F 46D2CEF3F502287B98272793F537F9788A4AA4F3F567942A85EE8713F58B490AA561F0 13F59E726EC532C0D3F59E726EC532C143F58B490AA561F073F567942A85EE87C3F537 F9788A4AA5E3F502287B98272883F4977F2F46D2D0F3F432BE24DCFB26A3F3B8730F23 188743F32D994700FA2E13F289F1B37A62F323F1EAC61E5BE60263F123916CA978FE43 F04A6C4282F0AF23EF652492CC90DDE3EE702D1069D2B8D3ED6A0075DC1FB0D3EC5379 90CA69AAD3EB2FA18D7EA6DCD3EA030413852C8123E8A57B3C1CB362A3E778FA957CF9 7693E8CF55BD9A57F883EA0F926605E253C3EB2FA18D7EA6D933EC43C78B462D9F53ED 494D42133EAC53EE3F6EFE811F0D43EF2788EC0A4F28D3F004C8555CEA8D53F0B6F47D 53C99483F1605961E6C888A3F20DC10E9606D623F289F1B37A62F0D3F3125AE183155D 73F36C78897C08E6B3F3CDCCECFBA86DF3F41708CB27C111F3F4419CBE8154FEF3F461 8C789182A933F472AFEBDDB7B5D3F472AFEBDDB7B633F4618C789182A973F4419CBE81 54FF83F41708CB27C112C3F3CDCCECFBA86FC3F36C78897C08E883F3125AE183155F13 F289F1B37A62F3E3F20DC10E9606D853F1605961E6C88AA3F0B6F47D53C997F3F004C8 555CEA8F53EF2788EC0A4F2BB3EE3F6EFE811F1103ED494D42133EB033EC43C78B462D A323EB2FA18D7EA6DCD3EA0F926605E25703E8CF55BD9A57FED3E778FA957CF97C73E6 4196091F447113E78B40CBDAE288A3E8CF55BD9A57F883EA030413852C7E23EB143428 C3835223EC18EA24E03DF173ED107F16005123F3EDF837FD84109C53EEBCED519B1B9F D3EF7674A05616C673F02C934FDEF17C23F0CC3BCF94D4AD33F1500EFB20266393F1D4 1552AECE9EF3F236EA7E44441513F289F1B37A62F183F2DC111BE40C6DF3F3125AE183 155E83F32D994700FA2D73F33C380AA9B7A633F33C380AA9B7A633F32D994700FA2E13 F3125AE183155F13F2DC111BE40C6FC3F289F1B37A62F323F236EA7E44441653F1D415 52AECEA0D3F1500EFB20266583F0CC3BCF94D4B0B3F02C934FDEF17E83EF7674A05616 C963EEBCED519B1BA433EDF837FD8410A143ED107F16005126A3EC18EA24E03DF433EB 143428C3835563EA030413852C8123E8CF55BD9A57FED3E78B40CBDAE28D63E6419609 1F447623E505A628C699FE33E64196091F447113E778FA957CF97693E8A57B3C1CB35D B3E9C1733B307332F3EAC91DA8CECDA833EBBB6ADC2AE25333EC9A3E23EEEF8153ED6A 0075DC1FAC93EE30AA2E62F586A3EEE91BD676FC78D3EF7674A05616C673F0116C9443 7469D3F07CD79B5647CE13F0F9EF70AEF97FB3F14085652F621BE3F183567921B49C03 F1BE7117DC636B03F1EAC61E5BE60163F201483FDC9F1243F201483FDC9F1243F1EAC6 1E5BE60263F1BE7117DC636BE3F183567921B49D83F14085652F621D23F0F9EF70AEF9 81D3F07CD79B5647CFA3F0116C9443746B73EF7674A05616C963EEE91BD676FC7C93EE 30AA2E62F58913ED6A0075DC1FB023EC9A3E23EEEF8543EBBB6ADC2AE257A3EAC91DA8 CECDACC3E9C1733B30733843E8A57B3C1CB362A3E778FA957CF97C73E64196091F4476 23E505A628C69A013-%&COLORG6&%$RGBG$\"\")!\"\"$\"\"'F6$\"\"\"F)-%*AXESS TYLEG6#%$BOXG-%+LIGHTMODELG6#%(LIGHT_4G-%+AXESLABELSG6%%\"xG%\"yG%\"uG " 1 2 0 1 10 0 2 1 6 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "u := 'u':\npde := x*diff(u(x,y),y)=y*diff(u(x,y),x);\npdsolve(pd e);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/*&%\"xG\"\"\"-%%diffG6$ -%\"uG6$F'%\"yGF/F(*&F/F(-F*6$F,F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%\"uG6$%\"xG%\"yG-%$_F1G6#,&*$)F'\"\"#\"\"\"F0*$)F(F/F0F0" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 9 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 43 "C onsider the partial differential equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),x,y)" "6#-%%DiffG6%-%\"uG6$%\"xG% \"yGF)F*" }{XPPEDIT 18 0 "``+2;" "6#,&%!G\"\"\"\"\"#F%" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Diff(u(x,y),x) = y;" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"y GF*F+" }{TEXT -1 12 " ------- (i)" }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[xy]+2*u[x] = y;" "6#/,&&%\"uG6#%#xyG\"\"\"*&\"\"#F)&F&6#%\"xGF)F)%\"yG" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "p=u[x]" "6#/%\" pG&%\"uG6#%\"xG" }{TEXT -1 12 ", that is, " }{XPPEDIT 18 0 "p(x,y)=Di ff(u(x,y),x)" "6#/-%\"pG6$%\"xG%\"yG-%%DiffG6$-%\"uG6$F'F(F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "Then equation (i) is " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(p(x,y),y)" "6#-% %DiffG6$-%\"pG6$%\"xG%\"yGF*" }{XPPEDIT 18 0 "``+2*p(x,y) = y;" "6#/,& %!G\"\"\"*&\"\"#F&-%\"pG6$%\"xG%\"yGF&F&F-" }{TEXT -1 15 " ------- (ii ). " }}{PARA 0 "" 0 "" {TEXT -1 5 "With " }{TEXT 320 1 "x" }{TEXT -1 85 " fixed, this equation is linear and can be solved by means of the \+ integrating factor " }{XPPEDIT 18 0 "exp(Int(2,y))=exp(2*y)" "6#/-%$ex pG6#-%$IntG6$\"\"#%\"yG-F%6#*&F*\"\"\"F+F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 51 "Multiplying (ii) by this integrating factor giv es: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(2*y)" "6# -%$expG6#*&\"\"#\"\"\"%\"yGF(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(p( x,y),y)" "6#-%%DiffG6$-%\"pG6$%\"xG%\"yGF*" }{XPPEDIT 18 0 "``+2*exp(2 *y) = y*exp(2*y);" "6#/,&%!G\"\"\"*&\"\"#F&-%$expG6#*&F(F&%\"yGF&F&F&* &F-F&-F*6#*&F(F&F-F&F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(exp (2*y)*p(x,y),y) = y*exp(2*y);" "6#/-%%DiffG6$*&-%$expG6#*&\"\"#\"\"\"% \"yGF-F--%\"pG6$%\"xGF.F-F.*&F.F--F)6#*&F,F-F.F-F-" }{TEXT -1 16 " --- ---- (iii). " }}{PARA 0 "" 0 "" {TEXT -1 17 "We can integrate " } {XPPEDIT 18 0 "y*exp(2*y)" "6#*&%\"yG\"\"\"-%$expG6#*&\"\"#F%F$F%F%" } {TEXT -1 17 " with respect to " }{TEXT 315 1 "y" }{TEXT -1 45 " using \+ the integration by parts in the form: " }{XPPEDIT 18 0 "Int(y*``(dv/dy ),y)=y*v-Int(v,y)" "6#/-%$IntG6$*&%\"yG\"\"\"-%!G6#*&%#dvGF)%#dyG!\"\" F)F(,&*&F(F)%\"vGF)F)-F%6$F3F(F0" }{TEXT -1 9 ", with " }{XPPEDIT 18 0 "v = exp(2*y)/2;" "6#/%\"vG*&-%$expG6#*&\"\"#\"\"\"%\"yGF+F+F*!\" \"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " } {XPPEDIT 18 0 "Int(y*exp(2*y),y) = y*exp(2*y)/2-Int(exp(2*y)/2,y);" "6 #/-%$IntG6$*&%\"yG\"\"\"-%$expG6#*&\"\"#F)F(F)F)F(,&*(F(F)-F+6#*&F.F)F (F)F)F.!\"\"F)-F%6$*&-F+6#*&F.F)F(F)F)F.F4F(F4" }{XPPEDIT 18 0 "`` = y *exp(2*y)/2-exp(2*y)/4+k;" "6#/%!G,(*(%\"yG\"\"\"-%$expG6#*&\"\"#F(F'F (F(F-!\"\"F(*&-F*6#*&F-F(F'F(F(\"\"%F.F.%\"kGF(" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 24 "Hence (ii) implies that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(2*y)*p(x,y) = y*exp(2*y)/2-exp (2*y)/4+h(x);" "6#/*&-%$expG6#*&\"\"#\"\"\"%\"yGF*F*-%\"pG6$%\"xGF+F*, (*(F+F*-F&6#*&F)F*F+F*F*F)!\"\"F**&-F&6#*&F)F*F+F*F*\"\"%F5F5-%\"hG6#F /F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 104 "where, in the cu rrent context, the \"constant of integration\" has been replaced by an arbitrary function " }{XPPEDIT 18 0 "h(x);" "6#-%\"hG6#%\"xG" }{TEXT -1 4 " of " }{TEXT 316 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p(x,y )=y/2-1/4+h(x)*exp(-2*y)" "6#/-%\"pG6$%\"xG%\"yG,(*&F(\"\"\"\"\"#!\"\" F+*&F+F+\"\"%F-F-*&-%\"hG6#F'F+-%$expG6#,$*&F,F+F(F+F-F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 34 "Now, recalling that by definitio n " }{XPPEDIT 18 0 "Diff(u(x,y),x)=p(x,y)" "6#/-%%DiffG6$-%\"uG6$%\"xG %\"yGF*-%\"pG6$F*F+" }{TEXT -1 13 ", we obtain: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),x) = y/2-1/4+h(x)*exp(-2*y) ;" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yGF*,(*&F+\"\"\"\"\"#!\"\"F.*&F.F.\" \"%F0F0*&-%\"hG6#F*F.-%$expG6#,$*&F/F.F+F.F0F.F." }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Treating \+ " }{TEXT 311 1 "y" }{TEXT -1 47 " as a constant and integrating with r espect to " }{TEXT 312 1 "x" }{TEXT -1 13 " we see that " }{XPPEDIT 18 0 "u(x,y)" "6#-%\"uG6$%\"xG%\"yG" }{TEXT -1 15 " has the form: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y)=x*y/2-x/4+f(x) *exp(-2*y)+g(y)" "6#/-%\"uG6$%\"xG%\"yG,**(F'\"\"\"F(F+\"\"#!\"\"F+*&F 'F+\"\"%F-F-*&-%\"fG6#F'F+-%$expG6#,$*&F,F+F(F+F-F+F+-%\"gG6#F(F+" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y) = x/4*(2*y-1)+f(x)*exp(-2*y)+g (y);" "6#/-%\"uG6$%\"xG%\"yG,(*(F'\"\"\"\"\"%!\"\",&*&\"\"#F+F(F+F+F+F -F+F+*&-%\"fG6#F'F+-%$expG6#,$*&F0F+F(F+F-F+F+-%\"gG6#F(F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 10 "where f '(" }{TEXT 314 1 "x" } {TEXT -1 1 ")" }{XPPEDIT 18 0 "``=h(x)" "6#/%!G-%\"hG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(y)" "6#-%\"gG6#%\"yG" }{TEXT -1 30 " is an arbitrary function of " }{TEXT 313 1 "y" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "u := 'u':\npde := diff(u(x,y),x,y)+2*diff(u(x,y),x)=y;\npdsolve(pd e);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/,&-%%diffG6%-%\"uG6$%\" xG%\"yGF-F.\"\"\"*&\"\"#F/-F(6$F*F-F/F/F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG,(-%$_F1G6#F(\"\"\"*&-%$expG6#,$*&\" \"#F-F(F-!\"\"F--%$_F2G6#F'F-F-*(\"\"%F5F'F-,&*&F4F-F(F-F-F-F5F-F-" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 64 "Find the general sol ution of the partial differential equation " }{XPPEDIT 18 0 "Diff(u(x ,y),`$`(x,2)) = 0;" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F*\"\"#\" \"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "u(x,y) = f(y)*x+g(y);" "6#/-%\"uG6$%\"xG%\"yG,&*&-%\"fG 6#F(\"\"\"F'F.F.-%\"gG6#F(F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "u := 'u':\np de := diff(u(x,y),x$2)=0;\npdsolve(pde);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/-%%diffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F,\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG,&*&-%$_F1G6#F(\"\" \"F'F.F.-%$_F2GF-F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_______ ______________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 64 "Find the general \+ solution of the partial differential equation " }{XPPEDIT 18 0 "Diff( u(x,y),`$`(x,2)) = u(x,y);" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F* \"\"#-F(6$F*F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y)=f(y)*exp(x)+g(y)*exp(-x)" "6#/-%\"uG6$%\" xG%\"yG,&*&-%\"fG6#F(\"\"\"-%$expG6#F'F.F.*&-%\"gG6#F(F.-F06#,$F'!\"\" F.F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "u := 'u':\npde := diff(u(x,y),x$2)=u(x,y) ;\npdsolve(pde);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/-%%diffG6$ -%\"uG6$%\"xG%\"yG-%\"$G6$F,\"\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"uG6$%\"xG%\"yG,&*&-%$_F1G6#F(\"\"\"-%$expG6#F'F.F.*&-%$_F2GF-F.- F06#,$F'!\"\"F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "________ _____________________________" }}{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 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 64 "Find the general \+ solution of the partial differential equation " }{XPPEDIT 18 0 "Diff( u(x,y),`$`(x,2))" "6#-%%DiffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F)\"\"#" } {XPPEDIT 18 0 "``+4*u(x,y) = 0;" "6#/,&%!G\"\"\"*&\"\"%F&-%\"uG6$%\"xG %\"yGF&F&\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y) = f(y)*cos*2*x+g(y)*sin*2*x;" "6#/-%\"uG6 $%\"xG%\"yG,&**-%\"fG6#F(\"\"\"%$cosGF.\"\"#F.F'F.F.**-%\"gG6#F(F.%$si nGF.F0F.F'F.F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "u := 'u':\npde := diff(u(x,y ),x$2)+4*u(x,y)=0;\npdsolve(pde);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %$pdeG/,&-%%diffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F-\"\"#\"\"\"*&\"\"%F3F*F 3F3\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG,&*&-%$ _F1G6#F(\"\"\"-%$sinG6#,$*&\"\"#F.F'F.F.F.F.*&-%$_F2GF-F.-%$cosGF1F.F. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "________________________ _____________" }}{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 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 64 "Find the general solution of the \+ partial differential equation " }{XPPEDIT 18 0 "Diff(u(x,y),`$`(x,2)) " "6#-%%DiffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F)\"\"#" }{XPPEDIT 18 0 "``+2 " "6#,&%!G\"\"\"\"\"#F%" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Diff(u(x,y), x);" "6#-%%DiffG6$-%\"uG6$%\"xG%\"yGF)" }{XPPEDIT 18 0 "``+2*u(x,y) = \+ 0;" "6#/,&%!G\"\"\"*&\"\"#F&-%\"uG6$%\"xG%\"yGF&F&\"\"!" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y) = f(y )*exp(-x)*sin*x+g(y)*exp(-x)*cos*x;" "6#/-%\"uG6$%\"xG%\"yG,&**-%\"fG6 #F(\"\"\"-%$expG6#,$F'!\"\"F.%$sinGF.F'F.F.**-%\"gG6#F(F.-F06#,$F'F3F. %$cosGF.F'F.F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "u := 'u':\npde := diff(u(x,y ),x$2)+2*diff(u(x,y),x)+2*u(x,y)=0;\npdsolve(pde);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$pdeG/,(-%%diffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F-\"\"# \"\"\"*&F2F3-F(6$F*F-F3F3*&F2F3F*F3F3\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG,&*(-%$_F1G6#F(\"\"\"-%$expG6#,$F'! \"\"F.-%$sinG6#F'F.F.*(-%$_F2GF-F.F/F.-%$cosGF6F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }} {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 37 "__ ___________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }} {PARA 0 "" 0 "" {TEXT -1 65 "Find the general solution of the partial \+ differential equation " }{XPPEDIT 18 0 "Diff(u(x,y),x,y) = Diff(u(x, y),x);" "6#/-%%DiffG6%-%\"uG6$%\"xG%\"yGF*F+-F%6$-F(6$F*F+F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x ,y) = exp(y)*f(x)+g(y);" "6#/-%\"uG6$%\"xG%\"yG,&*&-%$expG6#F(\"\"\"-% \"fG6#F'F.F.-%\"gG6#F(F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "u := 'u':\npde := dif f(u(x,y),x,y)=diff(u(x,y),x);\npdsolve(pde);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/-%%diffG6%-%\"uG6$%\"xG%\"yGF,F--F'6$F)F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG,&-%$_F1G6#F(\"\"\" *&-%$expGF,F--%$_F2G6#F'F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{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 37 "_________________________________ ____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 65 "Find the gen eral solution of the partial differential equation " }{XPPEDIT 18 0 "Diff(u(x,y),x,y)" "6#-%%DiffG6%-%\"uG6$%\"xG%\"yGF)F*" }{XPPEDIT 18 0 "``+2;" "6#,&%!G\"\"\"\"\"#F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff( u(x,y),x) = x;" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yGF*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 " Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y)=x^2/4+ex p(-2*y)*f(x)+g(y)" "6#/-%\"uG6$%\"xG%\"yG,(*&F'\"\"#\"\"%!\"\"\"\"\"*& -%$expG6#,$*&F+F.F(F.F-F.-%\"fG6#F'F.F.-%\"gG6#F(F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "u := 'u':\npde := diff(u(x,y),x,y)+2*diff(u(x,y),x)=x;\npdsolve( pde);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/,&-%%diffG6%-%\"uG6$% \"xG%\"yGF-F.\"\"\"*&\"\"#F/-F(6$F*F-F/F/F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG,(-%$_F1G6#F(\"\"\"*&-%$expG6#,$*&\" \"#F-F(F-!\"\"F--%$_F2G6#F'F-F-*&\"\"%F5F'F4F-" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{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 37 "____________ _________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q7 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Construct a partial diffe rential equation which has the general solution " }{XPPEDIT 18 0 "u(x, y)=f(x*y)" "6#/-%\"uG6$%\"xG%\"yG-%\"fG6#*&F'\"\"\"F(F-" }{TEXT -1 8 " , where " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 49 " is an arbitrary function of a single variable. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 318 1 "x" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),x)=y" "6#/-%%DiffG6$-%\"uG6 $%\"xG%\"yGF*F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),y);" "6#- %%DiffG6$-%\"uG6$%\"xG%\"yGF*" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "x* u[x]=y*u[y]" "6#/*&%\"xG\"\"\"&%\"uG6#F%F&*&%\"yGF&&F(6#F+F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "u := 'u':\npde := x*diff(u(x,y),x)=y*diff(u(x,y),y); \npdsolve(pde);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/*&%\"xG\"\" \"-%%diffG6$-%\"uG6$F'%\"yGF'F(*&F/F(-F*6$F,F/F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG-%$_F1G6#*&F'\"\"\"F(F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "__ ___________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q8 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Construct a partial differential equation which has the general solution " } {XPPEDIT 18 0 "u(x,y) = f(x/y);" "6#/-%\"uG6$%\"xG%\"yG-%\"fG6#*&F'\" \"\"F(!\"\"" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#% \"tG" }{TEXT -1 49 " is an arbitrary function of a single variable. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 319 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),x);" "6#-%%DiffG6$-%\"uG6$%\"xG%\"yGF)" }{XPPEDIT 18 0 "``+y" "6#,&%!G\"\" \"%\"yGF%" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Diff(u(x,y),y) = 0;" "6#/- %%DiffG6$-%\"uG6$%\"xG%\"yGF+\"\"!" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "x*u[x]+y*u[y] = 0;" "6#/,&*&%\"xG\"\"\"&%\"uG6#F&F'F'*&%\"yGF'&F)6# F,F'F'\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "u := 'u':\npde := x*diff(u(x ,y),x)+y*diff(u(x,y),y)=0;\npdsolve(pde);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/,&*&%\"xG\"\"\"-%%diffG6$-%\"uG6$F(%\"yGF(F)F)* &F0F)-F+6$F-F0F)F)\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$% \"xG%\"yG-%$_F1G6#*&F(\"\"\"F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 37 "_____________________________________" }}{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 37 "____________________ _________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q9 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Construct a partial differential equ ation which has the general solution " }{XPPEDIT 18 0 "u(x,y) = x*f(x, y);" "6#/-%\"uG6$%\"xG%\"yG*&F'\"\"\"-%\"fG6$F'F(F*" }{TEXT -1 8 ", wh ere " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 49 " is an arb itrary function of a single variable. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 317 1 "x" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),x);" "6#-%%DiffG6$-%\"uG6$% \"xG%\"yGF)" }{XPPEDIT 18 0 "``-y;" "6#,&%!G\"\"\"%\"yG!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Diff(u(x,y),y) = u;" "6#/-%%DiffG6$-%\"uG6$% \"xG%\"yGF+F(" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "x*u[x]-y*u[y] = u; " "6#/,&*&%\"xG\"\"\"&%\"uG6#F&F'F'*&%\"yGF'&F)6#F,F'!\"\"F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "u := 'u':\npde := x*diff(u(x,y),x)-y*diff(u(x,y),y)=u (x,y);\npdsolve(pde);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/,&*&% \"xG\"\"\"-%%diffG6$-%\"uG6$F(%\"yGF(F)F)*&F0F)-F+6$F-F0F)!\"\"F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG*&-%$_F1G6#*&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 37 "______________ _______________________" }}{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 37 "_____________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 4 "Q10 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Construct a partial differential equation which has the g eneral solution " }{XPPEDIT 18 0 "u(x,y) = f(x+2*y)+g(x-2*y);" "6#/-% \"uG6$%\"xG%\"yG,&-%\"fG6#,&F'\"\"\"*&\"\"#F.F(F.F.F.-%\"gG6#,&F'F.*&F 0F.F(F.!\"\"F." }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG 6#%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(t)" "6#-%\"gG6#%\"tG" } {TEXT -1 48 " are arbitrary functions of a single variable. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "An s " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),y$2)= 4" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F+\"\"#\"\"%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),x$2)" "6#-%%DiffG6$-%\"uG6$%\"xG%\"yG-% \"$G6$F)\"\"#" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "u[yy]=4*u[xx]" "6# /&%\"uG6#%#yyG*&\"\"%\"\"\"&F%6#%#xxGF*" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "u := \+ 'u':\npde := diff(u(x,y),y$2)=4*diff(u(x,y),x$2);\npdsolve(pde);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/-%%diffG6$-%\"uG6$%\"xG%\"yG-% \"$G6$F-\"\"#,$*&\"\"%\"\"\"-F'6$F)-F/6$F,F1F5F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG,&-%$_F1G6#,&*&\"\"#\"\"\"F(F0F0F'F0 F0-%$_F2G6#,&*&F/F0F(F0F0F'!\"\"F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 37 "_____________________________________" }}{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 37 "____________________ _________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 34 "Solutions for Laplace's equation " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "pde := diff(u(x,y),x$2)+diff(u(x,y),y$2)=0;\npdsolve(pde);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/,&-%%diffG6$-%\"uG6$%\"xG%\"yG -%\"$G6$F-\"\"#\"\"\"-F(6$F*-F06$F.F2F3\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG,&-%$_F1G6#,&F(\"\"\"*&F'F.^#F.F.F.F .-%$_F2G6#,&F(F.*&^#!\"\"F.F'F.F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "f := x -> exp(-x);\ng := x -> exp(-x);\nf(y+x*I)+g(y-I*x);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$expG6#,$9$! \"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)op eratorG%&arrowGF(-%$expG6#,$9$!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#,&%\"yG!\"\"*&^#F)\"\"\"%\"xGF,F,F,-F%6#,&F( F)*&F-F,^#F,F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"- %$expG6#,$%\"yG!\"\"F&-%$cosG6#%\"xGF&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }