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0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 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 }{PSTYLE "Bullet Item" -1 15 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 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Norm al" -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 79 "Finding solutions to partial diff erential equations by separating the variables" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 27.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exa mples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 167 "The technique of separating the variables which was u sed to find solutions for the wave equation can be used to find soluti ons of other partial differential equations." }}{PARA 0 "" 0 "" {TEXT -1 40 "The following examples illustrate this. " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 263 8 "Question" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 54 "Find solutions for the partial differenti al equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 264 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$%\"xG%\"yGF*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "by separating the variables. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 " " 0 "" {TEXT -1 13 "Suppose that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 266 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),x)=y" "6# /-%%DiffG6$-%\"uG6$%\"xG%\"yGF*F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Dif f(u(x,y),y)" "6#-%%DiffG6$-%\"uG6$%\"xG%\"yGF*" }{TEXT -1 12 " ------- (i)" }}{PARA 0 "" 0 "" {TEXT -1 28 "has a solution of the form " } {XPPEDIT 18 0 "u(x,y)=v(x)*`.`*w(y)" "6#/-%\"uG6$%\"xG%\"yG*(-%\"vG6#F '\"\"\"%\".GF--%\"wG6#F(F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u (x,y),x) = v*`'`(x)*`.`*w(y);" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yGF***%\" vG\"\"\"-%\"'G6#F*F.%\".GF.-%\"wG6#F+F." }{TEXT -1 13 " ------- (ii)" }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Diff(u(x,y),y)=v(x)*`.`*w*`'`(y)" "6#/-%%DiffG6$-%\" uG6$%\"xG%\"yGF+**-%\"vG6#F*\"\"\"%\".GF0%\"wGF0-%\"'G6#F+F0" }{TEXT -1 16 " ------- (iii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Substituting for " }{XPPEDIT 18 0 "Diff(u(x,y),x)" " 6#-%%DiffG6$-%\"uG6$%\"xG%\"yGF)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 " Diff(u(x,y),y)" "6#-%%DiffG6$-%\"uG6$%\"xG%\"yGF*" }{TEXT -1 35 " from (ii) and (iii) in (i) gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x*v*`'`(x)*w(y) = y*v(x)*w*`'`(y);" "6#/**%\"xG\"\"\"% \"vGF&-%\"'G6#F%F&-%\"wG6#%\"yGF&**F.F&-F'6#F%F&F,F&-F)6#F.F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 57 "The variables can be separate d by dividing both sides by " }{XPPEDIT 18 0 "v(x)*w(y)" "6#*&-%\"vG6# %\"xG\"\"\"-%\"wG6#%\"yGF(" }{TEXT -1 10 " to give: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 274 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "v *`'`(x)/v(x) = y" "6#/*(%\"vG\"\"\"-%\"'G6#%\"xGF&-F%6#F*!\"\"%\"yG" } {TEXT -1 1 " " }{XPPEDIT 18 0 "w*`'`(y)/w(y)" "6#*(%\"wG\"\"\"-%\"'G6# %\"yGF%-F$6#F)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "The left side of this equation only dep ends on " }{TEXT 267 1 "x" }{TEXT -1 36 " and the right side only depe nds on " }{TEXT 268 1 "y" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 275 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "v*`'`(x)/v(x) = y" "6#/*(%\"vG\"\"\"-%\"'G6#% \"xGF&-F%6#F*!\"\"%\"yG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "w*`'`(y)/w(y) = k;" "6#/*(%\"wG\"\"\"-%\"'G6#%\"yGF&-F%6#F*!\"\"%\"kG" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 269 1 "k" }{TEXT -1 16 " is a constant. " }}{PARA 0 "" 0 "" {TEXT -1 50 "Thus we have t wo ordinary differential equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v*`'`(x)/v(x)=k/x" "6#/*(%\"vG\"\"\"-%\"'G6#%\"xGF& -F%6#F*!\"\"*&%\"kGF&F*F-" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/v" "6# *&\"\"\"F$%\"vG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dv/dx=k/x" "6#/* &%#dvG\"\"\"%#dxG!\"\"*&%\"kGF&%\"xGF(" }{TEXT -1 14 " ------- (iv) " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "w*`'`(y)/w(y)=k/y" "6#/*(%\"wG\"\"\"-%\"'G6#%\"yGF&- F%6#F*!\"\"*&%\"kGF&F*F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/w" "6#*& \"\"\"F$%\"wG!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "dw/dy = k/y;" "6# /*&%#dwG\"\"\"%#dyG!\"\"*&%\"kGF&%\"yGF(" }{TEXT -1 14 " ------- (v). \+ " }}{PARA 0 "" 0 "" {TEXT -1 21 "From (iv) we obtain: " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/v,v)=Int(k/x,x)" "6#/-%$In tG6$*&\"\"\"F(%\"vG!\"\"F)-F%6$*&%\"kGF(%\"xGF*F/" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(abs(v)) = k*ln(abs(x))+c;" "6#/-%#lnG6#-%$absG6# %\"vG,&*&%\"kG\"\"\"-F%6#-F(6#%\"xGF.F.%\"cGF." }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v = A*exp(ln(abs(x)^k));" "6#/%\"vG*&%\"AG\"\"\"-%$exp G6#-%#lnG6#)-%$absG6#%\"xG%\"kGF'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "A=``" "6#/%\"AG%!G" }{TEXT 270 1 " +" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(c)" "6#-%$expG6#%\"cG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v = A*abs(x)^k;" "6#/%\"vG*&%\"AG\"\"\" )-%$absG6#%\"xG%\"kGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "One can check that simpler form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v = ``" "6#/%\"vG%!G" }{XPPEDIT 18 0 "v(x) = A*x^k; " "6#/-%\"vG6#%\"xG*&%\"AG\"\"\")F'%\"kGF*" }{TEXT -1 15 " ------- (vi ), " }}{PARA 257 "" 0 "" {TEXT -1 23 "is a solution of (iv). " }} {PARA 0 "" 0 "" {TEXT -1 54 "Alternatively, since (iv) can be arranged in the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dv/d x-k*v/x=0" "6#/,&*&%#dvG\"\"\"%#dxG!\"\"F'*(%\"kGF'%\"vGF'%\"xGF)F)\" \"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 51 "we see that it is a linear diffferential equation. " }}{PARA 0 "" 0 "" {TEXT -1 37 "An \+ appropriate integrating factor is " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "exp(Int(-k/x,x))=exp(-k*ln(x))" "6#/-%$expG6#-%$IntG 6$,$*&%\"kG\"\"\"%\"xG!\"\"F/F.-F%6#,$*&F,F--%#lnG6#F.F-F/" }{XPPEDIT 18 0 "``=x^(-k)" "6#/%!G)%\"xG,$%\"kG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 53 "Writing the linear differential equation in the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^(-k)" "6# )%\"xG,$%\"kG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dv/dx-k*x^(-k-1)*v =0" "6#/,&*&%#dvG\"\"\"%#dxG!\"\"F'*(%\"kGF')%\"xG,&F+F)F'F)F'%\"vGF'F )\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "we see that " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\" \"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[ x^(-k)*v]=0" "6#/7#*&) %\"xG,$%\"kG!\"\"\"\"\"%\"vGF+\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^(-k)*v=A" "6#/*&)%\"xG,$%\"kG!\"\"\"\"\"%\"vGF*%\"AG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 318 1 "A" } {TEXT -1 15 " is a constant." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v=``" "6#/%\"vG%!G" } {XPPEDIT 18 0 "v(x)=A*x^k" "6#/-%\"vG6#%\"xG*&%\"AG\"\"\")F'%\"kGF*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 38 "In a similar way, equat ion (v) gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "w= `` " "6#/%\"wG%!G" }{XPPEDIT 18 0 "w(y) = B*y^k;" "6#/-%\"wG6#%\"yG*&% \"BG\"\"\")F'%\"kGF*" }{TEXT -1 16 " ------- (vii), " }}{PARA 0 "" 0 " " {TEXT -1 6 "where " }{TEXT 271 1 "B" }{TEXT -1 16 " is a constant. \+ " }}{PARA 0 "" 0 "" {TEXT -1 48 "Thus from (vi) and (vii) we obtain th e solution " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y) =``" "6#/-%\"uG6$%\"xG%\"yG%!G" }{XPPEDIT 18 0 "v(x)*w(y) = C*x^k*y^k; " "6#/*&-%\"vG6#%\"xG\"\"\"-%\"wG6#%\"yGF)*(%\"CGF))F(%\"kGF))F-F1F)" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 272 1 " C" }{TEXT -1 19 " is a constant, or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y) = C*(x*y)^k;" "6#/-%\"uG6$%\"xG%\"yG*&%\"CG \"\"\")*&F'F+F(F+%\"kGF+" }{TEXT -1 15 " ------- (viii)" }}{PARA 0 "" 0 "" {TEXT -1 42 "for the partial differential equation (i)." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 119 "Since sums of solutions of t he form (viii) are also solutions of (i), we can obtain many other sol utions. For example, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y)=4/(x*y)-sqrt(x*y)-7*x^3*y^3" "6#/-%\"uG6$%\"xG%\"yG,(*&\" \"%\"\"\"*&F'F,F(F,!\"\"F,-%%sqrtG6#*&F'F,F(F,F.*(\"\"(F,*$F'\"\"$F,F( F6F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "is a solution of \+ (i). " }}{PARA 0 "" 0 "" {TEXT -1 64 "This suggests that the general s olution of (i) is likely to be: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "u(x,y)=f(x*y)" "6#/-%\"uG6$%\"xG%\"yG-%\"fG6#*&F'\"\"\" F(F-" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 66 " is any arbitrary (differe ntiable) function of a single variable. " }}{PARA 0 "" 0 "" {TEXT -1 54 "This can easily be checked by partial differentiation." }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "u(x,y)=f(x*y)" "6#/-%\"uG6 $%\"xG%\"yG-%\"fG6#*&F'\"\"\"F(F-" }{TEXT -1 18 ", it follows that " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "diff(u(x,y),x)=y" "6 #/-%%diffG6$-%\"uG6$%\"xG%\"yGF*F+" }{TEXT -1 6 " f '( " }{XPPEDIT 18 0 "x*y" "6#*&%\"xG\"\"\"%\"yGF%" }{TEXT -1 4 " ), " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "diff(u(x,y),y)=x" "6#/-%%diffG6$-%\"u G6$%\"xG%\"yGF+F*" }{TEXT -1 6 " f '( " }{XPPEDIT 18 0 "x*y" "6#*&%\"x G\"\"\"%\"yGF%" }{TEXT -1 3 " )," }}{PARA 0 "" 0 "" {TEXT -1 3 "so " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 273 1 "x" }{TEXT -1 1 " " } {XPPEDIT 18 0 "diff(u(x,y),x) = x*y" "6#/-%%diffG6$-%\"uG6$%\"xG%\"yGF **&F*\"\"\"F+F-" }{TEXT -1 6 " f '( " }{XPPEDIT 18 0 "x*y" "6#*&%\"xG \"\"\"%\"yGF%" }{TEXT -1 2 " )" }{XPPEDIT 18 0 "``= y" "6#/%!G%\"yG" } {TEXT -1 1 " " }{XPPEDIT 18 0 "diff(u(x,y),y)" "6#-%%diffG6$-%\"uG6$% \"xG%\"yGF*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "pde := x*Diff(u(x,y),x)=y*Di ff(u(x,y),y);\npdsolve(pde,HINT=`*`):\nPDEtools[build](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/*&%\"xG\"\"\"-%%DiffG6$-%\"uG6$F'%\" yGF'F(*&F/F(-F*6$F,F/F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$% \"xG%\"yG**%$_C1G\"\"\")F'&%#_cG6#F+F+%$_C2GF+)F(F-F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "pde := \+ 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-" }}}{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 308 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 54 "Find solutions for the partial differential equation " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Diff(u(x,y),x) " "6#-%%DiffG6$-%\"uG6$%\"xG%\"yGF)" }{XPPEDIT 18 0 "`` +``" "6#,&%!G \"\"\"F$F%" }{XPPEDIT 18 0 "Diff(u(x,y),y)=2*(x+y)*u(x,y)" "6#/-%%Diff G6$-%\"uG6$%\"xG%\"yGF+*(\"\"#\"\"\",&F*F.F+F.F.-F(6$F*F+F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "by separating the variables. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 309 8 "Solut ion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Diff(u(x,y),x)" "6# -%%DiffG6$-%\"uG6$%\"xG%\"yGF)" }{XPPEDIT 18 0 "`` +``" "6#,&%!G\"\"\" F$F%" }{XPPEDIT 18 0 "Diff(u(x,y),y)=2*(x+y)*u(x,y)" "6#/-%%DiffG6$-% \"uG6$%\"xG%\"yGF+*(\"\"#\"\"\",&F*F.F+F.F.-F(6$F*F+F." }{TEXT -1 13 " ------- (i) " }}{PARA 0 "" 0 "" {TEXT -1 28 "has a solution of the fo rm " }{XPPEDIT 18 0 "u(x,y)=v(x)*`.`*w(y)" "6#/-%\"uG6$%\"xG%\"yG*(-% \"vG6#F'\"\"\"%\".GF--%\"wG6#F(F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " Diff(u(x,y),x)=v*`'`(x)*`.`*w(y)" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yGF*** %\"vG\"\"\"-%\"'G6#F*F.%\".GF.-%\"wG6#F+F." }{TEXT -1 13 " ------- (ii )" }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),y)=v(x)*`.`*w*`'`(y)" "6#/-%%DiffG6$-% \"uG6$%\"xG%\"yGF+**-%\"vG6#F*\"\"\"%\".GF0%\"wGF0-%\"'G6#F+F0" } {TEXT -1 16 " ------- (iii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 17 "Substituting for " }{XPPEDIT 18 0 "Diff(u (x,y),x)" "6#-%%DiffG6$-%\"uG6$%\"xG%\"yGF)" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "Diff(u(x,y),y)" "6#-%%DiffG6$-%\"uG6$%\"xG%\"yGF*" } {TEXT -1 35 " from (ii) and (iii) in (i) gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v*`'`(x)*w(y)+v(x)*w*`'`(y) = 2*(x+y)*v (x)*w(y);" "6#/,&*(%\"vG\"\"\"-%\"'G6#%\"xGF'-%\"wG6#%\"yGF'F'*(-F&6#F +F'F-F'-F)6#F/F'F'**\"\"#F',&F+F'F/F'F'-F&6#F+F'-F-6#F/F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "Dividing both sides by " } {XPPEDIT 18 0 "v(x)*w(y)" "6#*&-%\"vG6#%\"xG\"\"\"-%\"wG6#%\"yGF(" } {TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v*`'`(x)/v(x)+w*`'`(y)/w(y)=2*x+2*y" "6#/,&*(%\"vG\"\"\"-%\"'G6#%\" xGF'-F&6#F+!\"\"F'*(%\"wGF'-F)6#%\"yGF'-F06#F3F.F',&*&\"\"#F'F+F'F'*&F 8F'F3F'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "The variabl es can be separated to give " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "v*`'`(x)/v(x)-2*x=2*y-w*`'`(y)/w(y)" "6#/,&*(%\"vG\"\" \"-%\"'G6#%\"xGF'-F&6#F+!\"\"F'*&\"\"#F'F+F'F.,&*&F0F'%\"yGF'F'*(%\"wG F'-F)6#F3F'-F56#F3F.F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "The left side of this equation only depends on " }{TEXT 310 1 "x" }{TEXT -1 36 " and the right side only depends on " }{TEXT 311 1 "y" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v*`'`(x)/v(x)-2*x=2*y-w*`'`(y)/w(y )" "6#/,&*(%\"vG\"\"\"-%\"'G6#%\"xGF'-F&6#F+!\"\"F'*&\"\"#F'F+F'F.,&*& F0F'%\"yGF'F'*(%\"wGF'-F)6#F3F'-F56#F3F.F." }{XPPEDIT 18 0 "``=k" "6#/ %!G%\"kG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {TEXT 312 1 "k" }{TEXT -1 16 " is a constant. " }}{PARA 0 "" 0 "" {TEXT -1 50 "Thus we have two ordinary differential equations: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v*`'`(x)/v(x) = 2*x+k ;" "6#/*(%\"vG\"\"\"-%\"'G6#%\"xGF&-F%6#F*!\"\",&*&\"\"#F&F*F&F&%\"kGF &" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/v" "6#*&\"\"\"F$%\"vG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dv/dx = 2*x+k;" "6#/*&%#dvG\"\"\"%#dxG! \"\",&*&\"\"#F&%\"xGF&F&%\"kGF&" }{TEXT -1 14 " ------- (iv) " }} {PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "w*`'`(y)/w(y) = 2*y-k;" "6#/*(%\"wG\"\"\"-%\"'G6#%\"yGF &-F%6#F*!\"\",&*&\"\"#F&F*F&F&%\"kGF-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/w" "6#*&\"\"\"F$%\"wG!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "dw/dy \+ = 2*y-k;" "6#/*&%#dwG\"\"\"%#dyG!\"\",&*&\"\"#F&%\"yGF&F&%\"kGF(" } {TEXT -1 14 " ------- (v). " }}{PARA 0 "" 0 "" {TEXT -1 21 "From (iv) \+ we obtain: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/ v,v) = Int(``(2*x+k),x);" "6#/-%$IntG6$*&\"\"\"F(%\"vG!\"\"F)-F%6$-%!G 6#,&*&\"\"#F(%\"xGF(F(%\"kGF(F3" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(abs(v)) = x^2+k*x+c;" "6#/-%#lnG6#-%$absG6#%\"vG,(*$%\"xG\"\"#\" \"\"*&%\"kGF/F-F/F/%\"cGF/" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v=`` \+ " "6#/%\"vG%!G" }{XPPEDIT 18 0 "v(x) = A*exp(x^2+k*x);" "6#/-%\"vG6#% \"xG*&%\"AG\"\"\"-%$expG6#,&*$F'\"\"#F**&%\"kGF*F'F*F*F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "A=``" "6#/% \"AG%!G" }{TEXT 313 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(c)" "6#- %$expG6#%\"cG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 38 "In a si milar way, equation (v) gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "w=`` " "6#/%\"wG%!G" }{XPPEDIT 18 0 "w(y) = B*exp(y^2-k *y);" "6#/-%\"wG6#%\"yG*&%\"BG\"\"\"-%$expG6#,&*$F'\"\"#F**&%\"kGF*F'F *!\"\"F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {TEXT 314 1 "B" }{TEXT -1 16 " is a constant. " }}{PARA 0 "" 0 "" {TEXT -1 28 "Thus we obtain the solution " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y)=`` " "6#/-%\"uG6$%\"xG%\"yG%!G" } {XPPEDIT 18 0 "v(x)*w(y) = C*exp(x^2+k*x+y^2-k*y);" "6#/*&-%\"vG6#%\"x G\"\"\"-%\"wG6#%\"yGF)*&%\"CGF)-%$expG6#,**$F(\"\"#F)*&%\"kGF)F(F)F)*$ F-F5F)*&F7F)F-F)!\"\"F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 315 1 "C" }{TEXT -1 19 " is a constant, or " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y) = C*exp(x^2+y^ 2+k*(x-y));" "6#/-%\"uG6$%\"xG%\"yG*&%\"CG\"\"\"-%$expG6#,(*$F'\"\"#F+ *$F(F1F+*&%\"kGF+,&F'F+F(!\"\"F+F+F+" }{TEXT -1 15 " ------- (vi), " } }{PARA 0 "" 0 "" {TEXT -1 42 "for the partial differential equation (i )." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "Not es" }{TEXT -1 2 ": " }}{PARA 15 "" 0 "" {TEXT -1 45 "The solution (vi) can be checked as follows. " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " } {XPPEDIT 18 0 "u(x,y) = C*exp(x^2+y^2+k*(x-y))" "6#/-%\"uG6$%\"xG%\"yG *&%\"CG\"\"\"-%$expG6#,(*$F'\"\"#F+*$F(F1F+*&%\"kGF+,&F'F+F(!\"\"F+F+F +" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "diff(u(x,y),x)=C*exp(x^2+y^2+k*(x-y))*(2*x+k)" "6#/-%%diffG6$-%\"uG6$ %\"xG%\"yGF**(%\"CG\"\"\"-%$expG6#,(*$F*\"\"#F.*$F+F4F.*&%\"kGF.,&F*F. F+!\"\"F.F.F.,&*&F4F.F*F.F.F7F.F." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "diff(u(x,y),x) = u(x,y)*(2*x+k);" "6#/-%%diffG6$-%\"uG6$%\"xG%\"yGF **&-F(6$F*F+\"\"\",&*&\"\"#F/F*F/F/%\"kGF/F/" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "diff(u(x,y),x) = 2*x*u(x,y)+k*u(x,y);" "6#/-%%diffG6$-% \"uG6$%\"xG%\"yGF*,&*(\"\"#\"\"\"F*F/-F(6$F*F+F/F/*&%\"kGF/-F(6$F*F+F/ F/" }{TEXT -1 16 " ------- (vii), " }}{PARA 0 "" 0 "" {TEXT -1 4 "and \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "diff(u(x,y),y)=C* exp(x^2+y^2+k*(x-y))*(2*y-k)" "6#/-%%diffG6$-%\"uG6$%\"xG%\"yGF+*(%\"C G\"\"\"-%$expG6#,(*$F*\"\"#F.*$F+F4F.*&%\"kGF.,&F*F.F+!\"\"F.F.F.,&*&F 4F.F+F.F.F7F9F." }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 9 "that is , " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "diff(u(x,y),y) \+ = u(x,y)*(2*y-k);" "6#/-%%diffG6$-%\"uG6$%\"xG%\"yGF+*&-F(6$F*F+\"\"\" ,&*&\"\"#F/F+F/F/%\"kG!\"\"F/" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "di ff(u(x,y),y) = 2*y*u(x,y)-k*u(x,y);" "6#/-%%diffG6$-%\"uG6$%\"xG%\"yGF +,&*(\"\"#\"\"\"F+F/-F(6$F*F+F/F/*&%\"kGF/-F(6$F*F+F/!\"\"" }{TEXT -1 17 " ------- (viii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "On adding equations (vii) and (viii) the terms involvi ng " }{TEXT 316 1 "k" }{TEXT -1 17 " cancel to give: " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Diff(u(x,y),x)" "6#-%%DiffG6$-% \"uG6$%\"xG%\"yGF)" }{XPPEDIT 18 0 "`` +``" "6#,&%!G\"\"\"F$F%" } {XPPEDIT 18 0 "Diff(u(x,y),y)=2*(x+y)*u(x,y)" "6#/-%%DiffG6$-%\"uG6$% \"xG%\"yGF+*(\"\"#\"\"\",&F*F.F+F.F.-F(6$F*F+F." }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 57 "which is the original partial differentia l equation (i). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 31 "The general solution of (i) is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y)=f(x-y)*exp(2*x*y)" "6#/-%\"uG6$% \"xG%\"yG*&-%\"fG6#,&F'\"\"\"F(!\"\"F.-%$expG6#*(\"\"#F.F'F.F(F.F." } {TEXT -1 15 " ------- (ix), " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 65 " is an arbitrary \+ (differentiable) function of a single variable. " }}{PARA 0 "" 0 "" {TEXT -1 53 "This can also be checked by partial differentiation. " }} {PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "u(x,y) = f(x-y)*ex p(2*x*y)" "6#/-%\"uG6$%\"xG%\"yG*&-%\"fG6#,&F'\"\"\"F(!\"\"F.-%$expG6# *(\"\"#F.F'F.F(F.F." }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "diff(u(x,y),x)=``" "6#/-%%diffG6$-%\"uG6$%\"xG%\"yG F*%!G" }{TEXT -1 3 "f '" }{XPPEDIT 18 0 "``( x-y )*exp(2*x*y) + f(x-y) *2*y*exp(2*x*y)" "6#,&*&-%!G6#,&%\"xG\"\"\"%\"yG!\"\"F*-%$expG6#*(\"\" #F*F)F*F+F*F*F***-%\"fG6#,&F)F*F+F,F*F1F*F+F*-F.6#*(F1F*F)F*F+F*F*F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "diff(u(x,y),x)=``" "6#/-%%diffG6$-%\" uG6$%\"xG%\"yGF*%!G" }{TEXT -1 3 "f '" }{XPPEDIT 18 0 "``(x-y)*exp(2*x *y)+2*y*u(x,y);" "6#,&*&-%!G6#,&%\"xG\"\"\"%\"yG!\"\"F*-%$expG6#*(\"\" #F*F)F*F+F*F*F**(F1F*F+F*-%\"uG6$F)F+F*F*" }{TEXT -1 14 " ------- (x), " }}{PARA 0 "" 0 "" {TEXT -1 6 "while " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "diff(u(x,y),y)=-``" "6#/-%%diffG6$-%\"uG6$%\"xG% \"yGF+,$%!G!\"\"" }{TEXT -1 3 "f '" }{XPPEDIT 18 0 "``(x-y)*exp(2*x*y) + f(x-y)*2*x*exp(2*x*y)" "6#,&*&-%!G6#,&%\"xG\"\"\"%\"yG!\"\"F*-%$exp G6#*(\"\"#F*F)F*F+F*F*F***-%\"fG6#,&F)F*F+F,F*F1F*F)F*-F.6#*(F1F*F)F*F +F*F*F*" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "diff(u(x,y),y)=-``" "6#/-%% diffG6$-%\"uG6$%\"xG%\"yGF+,$%!G!\"\"" }{TEXT -1 3 "f '" }{XPPEDIT 18 0 "``(x-y)*exp(2*x*y)+2*x*u(x,y);" "6#,&*&-%!G6#,&%\"xG\"\"\"%\"yG!\" \"F*-%$expG6#*(\"\"#F*F)F*F+F*F*F**(F1F*F)F*-%\"uG6$F)F+F*F*" }{TEXT -1 15 " ------- (xi). " }}{PARA 0 "" 0 "" {TEXT -1 27 "Adding (x) and \+ (xi) gives: " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Dif f(u(x,y),x)" "6#-%%DiffG6$-%\"uG6$%\"xG%\"yGF)" }{XPPEDIT 18 0 "`` +`` " "6#,&%!G\"\"\"F$F%" }{XPPEDIT 18 0 "Diff(u(x,y),y)=2*(x+y)*u(x,y)" " 6#/-%%DiffG6$-%\"uG6$%\"xG%\"yGF+*(\"\"#\"\"\",&F*F.F+F.F.-F(6$F*F+F. " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 57 "which is the original partial differential equation (i). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 18 "The solution (vi) " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y) = C*exp(x^2+y^2+k*(x-y));" "6# /-%\"uG6$%\"xG%\"yG*&%\"CG\"\"\"-%$expG6#,(*$F'\"\"#F+*$F(F1F+*&%\"kGF +,&F'F+F(!\"\"F+F+F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 104 "found by separating the variables can be seen to have the form of the general solution (ix) as follows. " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "u(x,y) = C*exp(x^2+y^2+k*(x-y))" "6#/-%\"uG6$%\"xG% \"yG*&%\"CG\"\"\"-%$expG6#,(*$F'\"\"#F+*$F(F1F+*&%\"kGF+,&F'F+F(!\"\"F +F+F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = C*exp(x^2-2*x*y+y^2+k*(x-y)+2*x*y);" "6#/%!G*&%\"CG\"\"\"-% $expG6#,,*$%\"xG\"\"#F'*(F.F'F-F'%\"yGF'!\"\"*$F0F.F'*&%\"kGF',&F-F'F0 F1F'F'*(F.F'F-F'F0F'F'F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = C*exp((x-y)^2+k*(x-y))*exp(2*x*y);" "6#/ %!G*(%\"CG\"\"\"-%$expG6#,&*$,&%\"xGF'%\"yG!\"\"\"\"#F'*&%\"kGF',&F.F' F/F0F'F'F'-F)6#*(F1F'F.F'F/F'F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = f(x-y)*exp(2*x*y);" "6#/%!G*&-%\"f G6#,&%\"xG\"\"\"%\"yG!\"\"F+-%$expG6#*(\"\"#F+F*F+F,F+F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "f(t)=C*exp( t^2+k*t)" "6#/-%\"fG6#%\"tG*&%\"CG\"\"\"-%$expG6#,&*$F'\"\"#F**&%\"kGF *F'F*F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "pde := Diff(u(x,y),x)+Diff(u(x,y),y)=2*(x+y)*u(x,y);\npdsolve(pde,HINT=`* `):\nPDEtools[build](%):\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/,&-%%DiffG6$-%\"uG6$%\"xG%\"yGF-\"\"\"-F(6$F*F.F/,$*(\" \"#F/,&F-F/F.F/F/F*F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$% \"xG%\"yG*(%$_C1G\"\"\"-%$expG6#,**&&%#_cG6#F+F+F'F+!\"\"*$)F'\"\"#F+F +*&F1F+F(F+F+*$)F(F7F+F+F+%$_C2GF+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "pde := Diff(u(x,y),x)+Diff( u(x,y),y)=2*(x+y)*u(x,y);\nsimplify(pdsolve(pde));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$pdeG/,&-%%DiffG6$-%\"uG6$%\"xG%\"yGF-\"\"\"-F(6$F* F.F/,$*(\"\"#F/,&F-F/F.F/F/F*F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"uG6$%\"xG%\"yG*&-%$_F1G6#,&F(\"\"\"F'!\"\"F.-%$expG6#,$*(\"\"#F.F( F.F'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 \+ 3 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 276 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 53 "Find solutions for the partial differential equation " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),x,y)" "6#-%%DiffG6%- %\"uG6$%\"xG%\"yGF)F*" }{XPPEDIT 18 0 "``+y;" "6#,&%!G\"\"\"%\"yGF%" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),y)=u(x,y)" "6#/-%%DiffG6$-% \"uG6$%\"xG%\"yGF+-F(6$F*F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "by separating the variables. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 277 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 " " 0 "" {TEXT -1 13 "Suppose that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),x,y)" "6#-%%DiffG6%-%\"uG6$%\"xG%\"yGF)F* " }{XPPEDIT 18 0 "``+y;" "6#,&%!G\"\"\"%\"yGF%" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Diff(u(x,y),y)=u(x,y)" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yG F+-F(6$F*F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 28 "has a solu tion of the form " }{XPPEDIT 18 0 "u(x,y)=v(x)*`.`*w(y)" "6#/-%\"uG6$ %\"xG%\"yG*(-%\"vG6#F'\"\"\"%\".GF--%\"wG6#F(F-" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),x,y) = v*`'`(x)*`.`*w*`'`(y);" "6#/-%%Diff G6%-%\"uG6$%\"xG%\"yGF*F+*,%\"vG\"\"\"-%\"'G6#F*F.%\".GF.%\"wGF.-F06#F +F." }{TEXT -1 13 " ------- (ii)" }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),y)=v(x) *`.`*w*`'`(y)" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yGF+**-%\"vG6#F*\"\"\"%\" .GF0%\"wGF0-%\"'G6#F+F0" }{TEXT -1 16 " ------- (iii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Substituting for " } {XPPEDIT 18 0 "Diff(u(x,y),x,y);" "6#-%%DiffG6%-%\"uG6$%\"xG%\"yGF)F* " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Diff(u(x,y),y)" "6#-%%DiffG6$-% \"uG6$%\"xG%\"yGF*" }{TEXT -1 35 " from (ii) and (iii) in (i) gives: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v*`'`(x)*w*`'`(y) +y*v(x)*w*`'`(y) = v(x)*w(y);" "6#/,&**%\"vG\"\"\"-%\"'G6#%\"xGF'%\"wG F'-F)6#%\"yGF'F'**F/F'-F&6#F+F'F,F'-F)6#F/F'F'*&-F&6#F+F'-F,6#F/F'" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "Dividing both sides by \+ " }{XPPEDIT 18 0 "v(x)*w*`'`(y);" "6#*(-%\"vG6#%\"xG\"\"\"%\"wGF(-%\"' G6#%\"yGF(" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v*`'`(x)/v(x)+y = w(y)/(w*`'`(y));" "6#/,&*(%\"vG\"\" \"-%\"'G6#%\"xGF'-F&6#F+!\"\"F'%\"yGF'*&-%\"wG6#F/F'*&F2F'-F)6#F/F'F. " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "The variables can be separated to give " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v*`'`(x)/v(x) = w(y)/(w*`'`(y))-y;" "6#/*(%\"vG\"\"\"-%\"'G6#%\"xGF &-F%6#F*!\"\",&*&-%\"wG6#%\"yGF&*&F1F&-F(6#F3F&F-F&F3F-" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 47 "The left side of this equation only depends on " }{TEXT 278 1 "x" }{TEXT -1 36 " and the right side only \+ depends on " }{TEXT 279 1 "y" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v*`'`(x)/v(x) = w(y)/(w*`'`(y))-y;" "6#/*(%\"vG\"\"\"-%\"'G6#%\"xGF&- F%6#F*!\"\",&*&-%\"wG6#%\"yGF&*&F1F&-F(6#F3F&F-F&F3F-" }{XPPEDIT 18 0 "``=k" "6#/%!G%\"kG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "wh ere " }{TEXT 280 1 "k" }{TEXT -1 16 " is a constant. " }}{PARA 0 "" 0 "" {TEXT -1 50 "Thus we have two ordinary differential equations: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v*`'`(x)/v(x) = k;" " 6#/*(%\"vG\"\"\"-%\"'G6#%\"xGF&-F%6#F*!\"\"%\"kG" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1/v" "6#*&\"\"\"F$%\"vG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dv/dx = k;" "6#/*&%#dvG\"\"\"%#dxG!\"\"%\"kG" }{TEXT -1 14 " ------- (iv) " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "w(y)/(w*`'`(y)) = y+k;" "6# /*&-%\"wG6#%\"yG\"\"\"*&F&F)-%\"'G6#F(F)!\"\",&F(F)%\"kGF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "w*`'`(y)/w(y)=1/(y+k)" "6#/*(%\"wG\"\" \"-%\"'G6#%\"yGF&-F%6#F*!\"\"*&F&F&,&F*F&%\"kGF&F-" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "dw/dy = w/(y+k);" "6#/*&%#dwG\"\"\"%#dyG!\"\"*&%\"wGF&, &%\"yGF&%\"kGF&F(" }{TEXT -1 14 " ------- (v). " }}{PARA 0 "" 0 "" {TEXT -1 21 "From (iv) we obtain: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(1/v,v) = Int(k,x);" "6#/-%$IntG6$*&\"\"\"F(%\"vG !\"\"F)-F%6$%\"kG%\"xG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(abs(v )) = k*x+c[1];" "6#/-%#lnG6#-%$absG6#%\"vG,&*&%\"kG\"\"\"%\"xGF.F.&%\" cG6#F.F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v=`` " "6#/%\"vG%!G" }{XPPEDIT 18 0 "v(x) = A*exp(k*x);" "6#/-%\"vG6#%\"xG*&%\"AG\"\"\"-%$e xpG6#*&%\"kGF*F'F*F*" }{TEXT -1 15 " ------- (vi), " }}{PARA 0 "" 0 " " {TEXT -1 6 "where " }{XPPEDIT 18 0 "A=``" "6#/%\"AG%!G" }{TEXT 281 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(c[1]);" "6#-%$expG6#&%\"cG6# \"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "From (v) we ob tain " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dw =(y+k) /w" "6#/*&%#dyG\"\"\"%#dwG!\"\"*&,&%\"yGF&%\"kGF&F&%\"wGF(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dw-y/w=k/w" "6#/,&*&%#dyG\"\"\"%#dwG !\"\"F'*&%\"yGF'%\"wGF)F)*&%\"kGF'F,F)" }{TEXT -1 16 " ------- (vii). \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "This \+ is a linear differential equation in the variable " }{TEXT 317 1 "y" } {TEXT -1 56 " which can be solved by means of the integrating factor \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(Int(-1/w,w))= exp(-ln(w))" "6#/-%$expG6#-%$IntG6$,$*&\"\"\"F,%\"wG!\"\"F.F--F%6#,$-% #lnG6#F-F." }{XPPEDIT 18 0 "``=1/w" "6#/%!G*&\"\"\"F&%\"wG!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 51 "Multiplying (vii) by th is integrating factor gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1/w" "6#*&\"\"\"F$%\"wG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dw-y/w^2=k/w^2" "6#/,&*&%#dyG\"\"\"%#dwG!\"\"F'*&%\" yGF'*$%\"wG\"\"#F)F)*&%\"kGF'*$F-F.F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "[y/w]=k/w^2" "6#/7#*&%\"yG\"\"\"%\"wG!\"\"*&%\"kGF'*$F( \"\"#F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "and then " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y/w=-k/w+c[2]" "6#/*& %\"yG\"\"\"%\"wG!\"\",&*&%\"kGF&F'F(F(&%\"cG6#\"\"#F&" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y+k=c[2]*w" "6#/,&%\"yG\"\"\"%\"kGF&*&& %\"cG6#\"\"#F&%\"wGF&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 " so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "w=`` " "6 #/%\"wG%!G" }{XPPEDIT 18 0 "w(y) = B*(y+k);" "6#/-%\"wG6#%\"yG*&%\"BG \"\"\",&F'F*%\"kGF*F*" }{TEXT -1 17 " ------- (viii), " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "B=1/c[2]" "6#/%\"BG*&\"\"\"F& &%\"cG6#\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 88 "Fr om (vi) and (viii) we see that the partial differential equation (i) h as the solution " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u (x,y) = C*exp(k*x)*(y+k);" "6#/-%\"uG6$%\"xG%\"yG*(%\"CG\"\"\"-%$expG6 #*&%\"kGF+F'F+F+,&F(F+F0F+F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 282 1 "C" }{TEXT -1 15 " is a constant." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "pde := Diff(u(x,y),x,y)+y*Diff(u(x,y),y)=u(x,y);\npdsolve(pde,HIN T=`*`):\nsimplify(PDEtools[build](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/,&-%%DiffG6%-%\"uG6$%\"xG%\"yGF-F.\"\"\"*&F.F/-F(6$F*F.F /F/F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG**%$_C1G\" \"\"-%$expG6#*&&%#_cG6#F+F+F'F+F+%$_C2GF+,&F(F+F0F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 295 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 39 "Find solutions for Lapl ace's equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "D iff(u(x,y),`$`(x,2))" "6#-%%DiffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F)\"\"#" }{XPPEDIT 18 0 "``+``" "6#,&%!G\"\"\"F$F%" }{XPPEDIT 18 0 "Diff(u(x,y) ,`$`(y,2))=0" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F+\"\"#\"\"!" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "by separating the variab les. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 296 8 " Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),`$`( x,2)) = -``;" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F*\"\"#,$%!G!\" \"" }{XPPEDIT 18 0 "Diff(u(x,y),`$`(y,2))" "6#-%%DiffG6$-%\"uG6$%\"xG% \"yG-%\"$G6$F*\"\"#" }{TEXT -1 13 " ------- (i) " }}{PARA 0 "" 0 "" {TEXT -1 28 "has a solution of the form " }{XPPEDIT 18 0 "u(x,y)=v(x) *`.`*w(y)" "6#/-%\"uG6$%\"xG%\"yG*(-%\"vG6#F'\"\"\"%\".GF--%\"wG6#F(F- " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),`$`(x,2)) = v*`''`(x )*`.`*w(y);" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F*\"\"#**%\"vG\" \"\"-%#''G6#F*F2%\".GF2-%\"wG6#F+F2" }{TEXT -1 13 " ------- (ii)" }} {PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Diff(u(x,y),`$`(y,2)) = v(x)*`.`*w*`''`(y);" "6#/-%%Di ffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F+\"\"#**-%\"vG6#F*\"\"\"%\".GF4%\"wGF4 -%#''G6#F+F4" }{TEXT -1 16 " ------- (iii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Substituting for " }{XPPEDIT 18 0 "Diff(u(x,y),`$`(x,2));" "6#-%%DiffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F )\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Diff(u(x,y),y)" "6#-%%Diff G6$-%\"uG6$%\"xG%\"yGF*" }{TEXT -1 35 " from (ii) and (iii) in (i) giv es: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v*`''`(x)*`.` *w(y) = -v(x)*`.`*w*`''`(y);" "6#/**%\"vG\"\"\"-%#''G6#%\"xGF&%\".GF&- %\"wG6#%\"yGF&,$**-F%6#F*F&F+F&F-F&-F(6#F/F&!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "The varia bles can be separated by dividing both sides by " }{XPPEDIT 18 0 "v(x) *w(y);" "6#*&-%\"vG6#%\"xG\"\"\"-%\"wG6#%\"yGF(" }{TEXT -1 10 " to giv e: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v*`''`(x)/v(x) = -w*`''`(y)/w(y)" "6#/*(%\"vG\"\"\"-%#''G6#%\"xGF&-F%6#F*!\"\",$*(% \"wGF&-F(6#%\"yGF&-F06#F3F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "The left side of this equation only depends on " }{TEXT 297 1 "x" }{TEXT -1 36 " and the right side only depends on " }{TEXT 298 1 "y" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v*`''`(x)/v(x) = -w*` ''`(y)/w(y)" "6#/*(%\"vG\"\"\"-%#''G6#%\"xGF&-F%6#F*!\"\",$*(%\"wGF&-F (6#%\"yGF&-F06#F3F-F-" }{XPPEDIT 18 0 "``=k" "6#/%!G%\"kG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 299 1 "k" }{TEXT -1 16 " is a constant. " }}{PARA 0 "" 0 "" {TEXT -1 50 "Thus we have t wo ordinary differential equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v*`''`(x)/v(x) = k;" "6#/*(%\"vG\"\"\"-%#''G6#%\"xG F&-F%6#F*!\"\"%\"kG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*v/(d*x^2) = \+ k*v;" "6#/*(%\"dG\"\"#%\"vG\"\"\"*&F%F(*$%\"xGF&F(!\"\"*&%\"kGF(F'F(" }{TEXT -1 14 " ------- (iv) " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "w*`''`(y)/w(y) = -k; " "6#/*(%\"wG\"\"\"-%#''G6#%\"yGF&-F%6#F*!\"\",$%\"kGF-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*w/(d*y^2) = -k*w;" "6#/*(%\"dG\"\"#%\"wG\"\"\"* &F%F(*$%\"yGF&F(!\"\",$*&%\"kGF(F'F(F," }{TEXT -1 14 " ------- (v). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " } {XPPEDIT 18 0 "k=0" "6#/%\"kG\"\"!" }{TEXT -1 76 ", the only solution \+ is the constant solution, so we may as well supose that " }{XPPEDIT 18 0 "k<>0" "6#0%\"kG\"\"!" }{TEXT -1 18 ". The cases where " }{TEXT 302 1 "k" }{TEXT -1 17 " is positive and " }{TEXT 301 1 "k" }{TEXT -1 47 " is negative need to be considered separately. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 6 "Case I" }{TEXT -1 3 ": \+ " }{TEXT 300 1 "k" }{TEXT -1 18 " is positive, say " }{XPPEDIT 18 0 " k=q^2" "6#/%\"kG*$%\"qG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "In this case the auxiliary equation " }{XPPEDIT 18 0 "m^ 2 = q^2;" "6#/*$%\"mG\"\"#*$%\"qGF&" }{TEXT -1 41 " of (iv) has the tw o real roots given by " }{XPPEDIT 18 0 "m=``" "6#/%\"mG%!G" }{TEXT 303 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "q;" "6#%\"qG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "Hence the general solution of (iv) i s" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v = c[1]*exp(q*x )+c[2]*exp(-q*x);" "6#/%\"vG,&*&&%\"cG6#\"\"\"F*-%$expG6#*&%\"qGF*%\"x GF*F*F**&&F(6#\"\"#F*-F,6#,$*&F/F*F0F*!\"\"F*F*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 24 "The auxiliary equation " }{XPPEDIT 18 0 "m^2 = -q^2;" "6#/*$%\"mG\"\"#,$*$%\"qGF&!\"\"" }{TEXT -1 50 " of (v) \+ has the two pure imaginary roots given by " }{XPPEDIT 18 0 "m=``" "6#/ %\"mG%!G" }{TEXT 304 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "q*i;" "6#*& %\"qG\"\"\"%\"iGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "He nce the general solution of (v) is " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "w = c[3]*cos*q*y+c[4]*sin*q*y;" "6#/%\"wG,&**&%\"cG6 #\"\"$\"\"\"%$cosGF+%\"qGF+%\"yGF+F+**&F(6#\"\"%F+%$sinGF+F-F+F.F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 "This gives the solutio n: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y) = (c[1] *exp(q*x)+c[2]*exp(-q*x))*(c[3]*cos*q*y+c[4]*sin*q*y);" "6#/-%\"uG6$% \"xG%\"yG*&,&*&&%\"cG6#\"\"\"F/-%$expG6#*&%\"qGF/F'F/F/F/*&&F-6#\"\"#F /-F16#,$*&F4F/F'F/!\"\"F/F/F/,&**&F-6#\"\"$F/%$cosGF/F4F/F(F/F/**&F-6# \"\"%F/%$sinGF/F4F/F(F/F/F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y) = exp(q*x)*(A*cos*q*y+B*sin*q*y)+exp(-q*x)*(P*cos*q*y+Q*sin*q*y);" "6#/ -%\"uG6$%\"xG%\"yG,&*&-%$expG6#*&%\"qG\"\"\"F'F0F0,&**%\"AGF0%$cosGF0F /F0F(F0F0**%\"BGF0%$sinGF0F/F0F(F0F0F0F0*&-F,6#,$*&F/F0F'F0!\"\"F0,&** %\"PGF0F4F0F/F0F(F0F0**%\"QGF0F7F0F/F0F(F0F0F0F0" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "for Lapla ce's equation, where " }{XPPEDIT 18 0 "A=c[1]*c[3]" "6#/%\"AG*&&%\"cG6 #\"\"\"F)&F'6#\"\"$F)" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "B = c[1]*c[3]; " "6#/%\"BG*&&%\"cG6#\"\"\"F)&F'6#\"\"$F)" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "P = c[2]*c[3];" "6#/%\"PG*&&%\"cG6#\"\"#\"\"\"&F'6#\"\"$F*" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "Q = c[2]*c[4];" "6#/%\"QG*&&%\"cG6# \"\"#\"\"\"&F'6#\"\"%F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 257 7 "Case II" }{TEXT -1 3 ": " }{TEXT 305 1 "k" }{TEXT -1 18 " is negat ive, say " }{XPPEDIT 18 0 "k = -q^2;" "6#/%\"kG,$*$%\"qG\"\"#!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 58 "A similar argument to t hat for case I gives the solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "u(x,y) = exp(q*y)*(A*cos*q*x+B*sin*q*x)+exp(-q*y)*(P *cos*q*x+Q*sin*q*x);" "6#/-%\"uG6$%\"xG%\"yG,&*&-%$expG6#*&%\"qG\"\"\" F(F0F0,&**%\"AGF0%$cosGF0F/F0F'F0F0**%\"BGF0%$sinGF0F/F0F'F0F0F0F0*&-F ,6#,$*&F/F0F(F0!\"\"F0,&**%\"PGF0F4F0F/F0F'F0F0**%\"QGF0F7F0F/F0F'F0F0 F0F0" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "A,B,P" "6%%\"AG%\"BG%\"PG" }{TEXT -1 5 " and " }{TEXT 306 1 "Q" }{TEXT -1 26 " are arbitrary constants. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "pde := Diff( u(x,y),x$2)+Diff(u(x,y),y$2)=0;\npdsolve(pde,HINT=`*`):\nPDEtools[buil d](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/,&-%%DiffG6$-%\"uG6$ %\"xG%\"yG-%\"$G6$F-\"\"#\"\"\"-F(6$F*-F06$F.F2F3\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG,***-%$expG6#*&&%#_cG6#\"\"\"#F 2\"\"#F'F2F2%$_C1GF2%$_C3GF2-%$sinG6#*&F/F3F(F2F2F2**F+F2F5F2%$_C4GF2- %$cosGF9F2F2**F+!\"\"%$_C2GF2F6F2F7F2F2**F+F@FAF2F " 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 283 8 "Question " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 54 "Find solutions for t he partial differential equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "a^2" "6#*$%\"aG\"\"#" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),x$2)=Diff(u(x,y),y)" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yG-% \"$G6$F*\"\"#-F%6$-F(6$F*F+F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "by separating the variables. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 284 8 "Solution" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a^2" "6#*$%\"aG\"\"#" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Diff(u(x,y),x$2)=Diff(u(x,y),y)" "6#/-%%DiffG6$-%\"uG6$ %\"xG%\"yG-%\"$G6$F*\"\"#-F%6$-F(6$F*F+F+" }{TEXT -1 13 " ------- (i) \+ " }}{PARA 0 "" 0 "" {TEXT -1 28 "has a solution of the form " } {XPPEDIT 18 0 "u(x,y)=v(x)*`.`*w(y)" "6#/-%\"uG6$%\"xG%\"yG*(-%\"vG6#F '\"\"\"%\".GF--%\"wG6#F(F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a^2" " 6#*$%\"aG\"\"#" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),`$`(x,2)) \+ = a^2*v*`''`(x)*`.`*w(y);" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F* \"\"#*,%\"aGF/%\"vG\"\"\"-%#''G6#F*F3%\".GF3-%\"wG6#F+F3" }{TEXT -1 13 " ------- (ii)" }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Diff(u(x,y),y)=v(x)*`.`*w*`'`(y) " "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yGF+**-%\"vG6#F*\"\"\"%\".GF0%\"wGF0-% \"'G6#F+F0" }{TEXT -1 16 " ------- (iii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Substituting for " }{XPPEDIT 18 0 "Diff(u(x,y),`$`(x,2));" "6#-%%DiffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F )\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Diff(u(x,y),y)" "6#-%%Diff G6$-%\"uG6$%\"xG%\"yGF*" }{TEXT -1 35 " from (ii) and (iii) in (i) giv es: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a^2*v*`''`(x) *`.`*w(y) = v(x)*`.`*w*`'`(y);" "6#/*,%\"aG\"\"#%\"vG\"\"\"-%#''G6#%\" xGF(%\".GF(-%\"wG6#%\"yGF(**-F'6#F,F(F-F(F/F(-%\"'G6#F1F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 57 "The variables can be separated b y dividing both sides by " }{XPPEDIT 18 0 "a^2*v(x)*w(y);" "6#*(%\"aG \"\"#-%\"vG6#%\"xG\"\"\"-%\"wG6#%\"yGF*" }{TEXT -1 10 " to give: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v*`''`(x)/v(x) = 1/(a ^2);" "6#/*(%\"vG\"\"\"-%#''G6#%\"xGF&-F%6#F*!\"\"*&F&F&*$%\"aG\"\"#F- " }{TEXT -1 2 " " }{XPPEDIT 18 0 "w*`'`(y)/w(y)" "6#*(%\"wG\"\"\"-%\" 'G6#%\"yGF%-F$6#F)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "The left side of this equation only depends on " }{TEXT 285 1 "x" }{TEXT -1 36 " and the right side only depends on " }{TEXT 286 1 "y" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v*`''`(x)/v(x) = 1/(a^2);" "6#/*(% \"vG\"\"\"-%#''G6#%\"xGF&-F%6#F*!\"\"*&F&F&*$%\"aG\"\"#F-" }{TEXT -1 2 " " }{XPPEDIT 18 0 "w*`'`(y)/w(y) = k;" "6#/*(%\"wG\"\"\"-%\"'G6#% \"yGF&-F%6#F*!\"\"%\"kG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 287 1 "k" }{TEXT -1 16 " is a constant. " }}{PARA 0 "" 0 "" {TEXT -1 50 "Thus we have two ordinary differential equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v*`''`(x)/v(x) = k;" "6#/*(%\"vG\"\"\"-%#''G6#%\"xGF&-F%6#F*!\"\"%\"kG" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*v/(d*x^2) = k*v;" "6#/*(%\"dG\"\"#%\"vG\"\"\"*& F%F(*$%\"xGF&F(!\"\"*&%\"kGF(F'F(" }{TEXT -1 14 " ------- (iv) " }} {PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "w*`'`(y)/w(y) = a^2*k;" "6#/*(%\"wG\"\"\"-%\"'G6#%\"yGF &-F%6#F*!\"\"*&%\"aG\"\"#%\"kGF&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/ w" "6#*&\"\"\"F$%\"wG!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "dw/dy = a ^2*k;" "6#/*&%#dwG\"\"\"%#dyG!\"\"*&%\"aG\"\"#%\"kGF&" }{TEXT -1 14 " \+ ------- (v). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "k=0" "6#/%\"kG\"\"!" }{TEXT -1 76 " , the only solution is the constant solution, so we may as well supose that " }{XPPEDIT 18 0 "k<>0" "6#0%\"kG\"\"!" }{TEXT -1 18 ". The case s where " }{TEXT 290 1 "k" }{TEXT -1 17 " is positive and " }{TEXT 289 1 "k" }{TEXT -1 47 " is negative need to be considered separately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 6 "Case I" }{TEXT -1 3 ": " }{TEXT 288 1 "k" }{TEXT -1 18 " is positive, say " }{XPPEDIT 18 0 "k=q^2" "6#/%\"kG*$%\"qG\"\"#" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 37 "In this case the auxiliary equation " } {XPPEDIT 18 0 "m^2 = q^2;" "6#/*$%\"mG\"\"#*$%\"qGF&" }{TEXT -1 41 " o f (iv) has the two real roots given by " }{XPPEDIT 18 0 "m=``" "6#/%\" mG%!G" }{TEXT 291 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "q;" "6#%\"qG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "Hence the general solu tion of (iv) is" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v \+ = c[1]*exp(q*x)+c[2]*exp(-q*x);" "6#/%\"vG,&*&&%\"cG6#\"\"\"F*-%$expG6 #*&%\"qGF*%\"xGF*F*F**&&F(6#\"\"#F*-F,6#,$*&F/F*F0F*!\"\"F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "The general solution of (v) i s " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "w = c[3]*exp(q^ 2*a^2*y);" "6#/%\"wG*&&%\"cG6#\"\"$\"\"\"-%$expG6#*(%\"qG\"\"#%\"aGF0% \"yGF*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 "This gives t he solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x, y) = c[1]*c[3]*exp(q*x)*exp(q^2*a^2*y)+c[2]*c[3]*exp(-q*x)*exp(q^2*a^2 *y);" "6#/-%\"uG6$%\"xG%\"yG,&**&%\"cG6#\"\"\"F.&F,6#\"\"$F.-%$expG6#* &%\"qGF.F'F.F.-F36#*(F6\"\"#%\"aGF:F(F.F.F.**&F,6#F:F.&F,6#F1F.-F36#,$ *&F6F.F'F.!\"\"F.-F36#*(F6F:F;F:F(F.F.F." }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y) = A*exp(q^2*a^2*y+q*x)+B*exp(q^2*a^2*y-q*x);" "6#/-%\"uG6 $%\"xG%\"yG,&*&%\"AG\"\"\"-%$expG6#,&*(%\"qG\"\"#%\"aGF3F(F,F,*&F2F,F' F,F,F,F,*&%\"BGF,-F.6#,&*(F2F3F4F3F(F,F,*&F2F,F'F,!\"\"F,F," }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "A=c[1] *c[3]" "6#/%\"AG*&&%\"cG6#\"\"\"F)&F'6#\"\"$F)" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "B=c[2]*c[3]" "6#/%\"BG*&&%\"cG6#\"\"#\"\"\"&F'6#\"\"$F* " }{TEXT -1 11 ", for (i). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 7 "Case II" }{TEXT -1 3 ": " }{TEXT 292 1 "k" } {TEXT -1 18 " is negative, say " }{XPPEDIT 18 0 "k=-q^2" "6#/%\"kG,$*$ %\"qG\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "In th is case the auxiliary equation " }{XPPEDIT 18 0 "m^2 = -q^2;" "6#/*$% \"mG\"\"#,$*$%\"qGF&!\"\"" }{TEXT -1 51 " of (iv) has the two pure ima ginary roots given by " }{XPPEDIT 18 0 "m=``" "6#/%\"mG%!G" }{TEXT 293 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "q*i;" "6#*&%\"qG\"\"\"%\"iGF %" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "Hence the general s olution of (iv) is" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v = c[1]*cos*q*x+c[2]*sin*q*x;" "6#/%\"vG,&**&%\"cG6#\"\"\"F*%$cosGF* %\"qGF*%\"xGF*F***&F(6#\"\"#F*%$sinGF*F,F*F-F*F*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 32 "The general solution of (v) is " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "w = c[3]*exp(-q^2*a^2 );" "6#/%\"wG*&&%\"cG6#\"\"$\"\"\"-%$expG6#,$*&%\"qG\"\"#%\"aGF1!\"\"F *" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 "This gives the solu tion: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y) = c[ 1]*c[3]*cos*q*x*exp(-q^2*a^2*y)+c[2]*c[3]*sin*q*x*exp(-q^2*a^2*y);" "6 #/-%\"uG6$%\"xG%\"yG,&*.&%\"cG6#\"\"\"F.&F,6#\"\"$F.%$cosGF.%\"qGF.F'F .-%$expG6#,$*(F3\"\"#%\"aGF9F(F.!\"\"F.F.*.&F,6#F9F.&F,6#F1F.%$sinGF.F 3F.F'F.-F56#,$*(F3F9F:F9F(F.F;F.F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u( x,y) = exp(-q^2*a^2*y)*(A*cos*q*x+B*sin*q*x);" "6#/-%\"uG6$%\"xG%\"yG* &-%$expG6#,$*(%\"qG\"\"#%\"aGF0F(\"\"\"!\"\"F2,&**%\"AGF2%$cosGF2F/F2F 'F2F2**%\"BGF2%$sinGF2F/F2F'F2F2F2" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 6 "where " }{XPPEDIT 18 0 "A=c[1]*c[3]" "6#/%\"AG*&&%\"cG6# \"\"\"F)&F'6#\"\"$F)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B=c[2]*c[3] " "6#/%\"BG*&&%\"cG6#\"\"#\"\"\"&F'6#\"\"$F*" }{TEXT -1 11 ", for (i). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 10 "Con clusion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 75 "The method of separation of variables leads to solutions of the two forms: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y) = A*exp(q^2*a^ 2*y+q*x)+B*exp(q^2*a^2*y-q*x);" "6#/-%\"uG6$%\"xG%\"yG,&*&%\"AG\"\"\"- %$expG6#,&*(%\"qG\"\"#%\"aGF3F(F,F,*&F2F,F'F,F,F,F,*&%\"BGF,-F.6#,&*(F 2F3F4F3F(F,F,*&F2F,F'F,!\"\"F,F," }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u (x,y) = exp(-q^2*a^2*y)*(A*cos*q*x+B*sin*q*x);" "6#/-%\"uG6$%\"xG%\"yG *&-%$expG6#,$*(%\"qG\"\"#%\"aGF0F(\"\"\"!\"\"F2,&**%\"AGF2%$cosGF2F/F2 F'F2F2**%\"BGF2%$sinGF2F/F2F'F2F2F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT 259 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 34 "T he partial differential equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a^2" "6#*$%\"aG\"\"#" }{TEXT -1 1 " " }{XPPEDIT 18 0 " Diff(u(x,t),`$`(x,2)) = Diff(u(x,t),t);" "6#/-%%DiffG6$-%\"uG6$%\"xG% \"tG-%\"$G6$F*\"\"#-F%6$-F(6$F*F+F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 8 "governs " }{TEXT 259 9 "heat flow" }{TEXT -1 47 " along \+ a straight thin wire oriented along the " }{TEXT 294 1 "x" }{TEXT -1 67 " axis and perfectly insulated to prevent heat loss from the sides. " }}{PARA 0 "" 0 "" {TEXT -1 80 "Useful solutions can be obtained fro m the second case considered above, namely: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,t)=exp(-q^2*a^2*t)*(A*cos*q*x+B*sin *q*x)" "6#/-%\"uG6$%\"xG%\"tG*&-%$expG6#,$*(%\"qG\"\"#%\"aGF0F(\"\"\"! \"\"F2,&**%\"AGF2%$cosGF2F/F2F'F2F2**%\"BGF2%$sinGF2F/F2F'F2F2F2" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "pde := a^2*Diff(u(x,y),x$2)=Diff(u(x,y),y);\npds olve(pde,HINT=`*`):\nPDEtools[build](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/*&)%\"aG\"\"#\"\"\"-%%DiffG6$-%\"uG6$%\"xG%\"yG-%\"$G6$F 1F)F*-F,6$F.F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG, &**-%$expG6#*&&%#_cG6#\"\"\"#F2\"\"#F'F2F2%$_C3GF2-F,6#*()%\"aGF4F2F/F 2F(F2F2%$_C1GF2F2**F+!\"\"F5F2F6F2%$_C2GF2F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 215 "pde := a^2* Diff(u(x,y),x$2)=Diff(u(x,y),y);\nsubs(u(x,y)=A*exp(q^2*a^2*y+q*x)+B*e xp(q^2*a^2*y-q*x),pde);\nsimplify(value(%));\nsubs(u(x,y)=A*exp(-q^2*a ^2*y)*cos(q*x)+B*exp(-q^2*a^2*y)*sin(q*x),pde);\nsimplify(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/*&)%\"aG\"\"#\"\"\"-%%DiffG6 $-%\"uG6$%\"xG%\"yG-%\"$G6$F1F)F*-F,6$F.F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%\"aG\"\"#\"\"\"-%%DiffG6$,&*&%\"AGF(-%$expG6#,&*() %\"qGF'F(F%F(%\"yGF(F(*&F5F(%\"xGF(F(F(F(*&%\"BGF(-F06#,&F3F(F7!\"\"F( F(-%\"$G6$F8F'F(-F*6$F,F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*()%\"aG \"\"#\"\"\")%\"qGF'F(,&*&%\"AGF(-%$expG6#*&F*F(,&*(%\"yGF(F%F(F*F(F(% \"xGF(F(F(F(*&%\"BGF(-F/6#*&F*F(,&F3F(F5!\"\"F(F(F(F(F$" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/*&)%\"aG\"\"#\"\"\"-%%DiffG6$,&*(%\"AGF(-%$expG 6#,$*()%\"qGF'F(F%F(%\"yGF(!\"\"F(-%$cosG6#*&F5F(%\"xGF(F(F(*(%\"BGF(F /F(-%$sinGF:F(F(-%\"$G6$F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 53 "Find solutions for t he partial differential equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),x);" "6#-%%DiffG6$-%\"uG6$%\"xG%\"yGF)" } {XPPEDIT 18 0 "``+`` " "6#,&%!G\"\"\"F$F%" }{XPPEDIT 18 0 "Diff(u(x,y) ,y) = 0;" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yGF+\"\"!" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 29 "by separating the variables. " }}{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) = A*exp(k*(x-y)) ;" "6#/-%\"uG6$%\"xG%\"yG*&%\"AG\"\"\"-%$expG6#*&%\"kGF+,&F'F+F(!\"\"F +F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "pde := Di ff(u(x,y),x)+Diff(u(x,y),y)=0;\npdsolve(pde,HINT=`*`):\ncombine(PDEtoo ls[build](%));\npdsolve(pde);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pd eG/,&-%%DiffG6$-%\"uG6$%\"xG%\"yGF-\"\"\"-F(6$F*F.F/\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG*(%$_C1G\"\"\"%$_C2GF+-%$exp G6#,&*&&%#_cG6#F+F+F'F+F+*&F2F+F(F+!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG-%$_F1G6#,&F(\"\"\"F'!\"\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{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 53 "Find solutions for the partial differenti al equation " }}{PARA 256 "" 0 "" {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 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "by separating the variab les. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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) = A*exp(k*x)*y^k;" "6#/-%\"uG6$%\"xG% \"yG*(%\"AG\"\"\"-%$expG6#*&%\"kGF+F'F+F+)F(F0F+" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "pde := Diff(u(x,y),x)=y*Diff (u(x,y),y);\npdsolve(pde,HINT=`*`):\nPDEtools[build](%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$pdeG/-%%DiffG6$-%\"uG6$%\"xG%\"yGF,*&F-\"\"\" -F'6$F)F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG**%$ _C1G\"\"\"-%$expG6#*&&%#_cG6#F+F+F'F+F+%$_C2GF+)F(F0F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {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 53 "Find solutions for the partial differenti al equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 307 1 "y" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),x) = x;" "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 29 "by separating the variables. " }}{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) = A*exp(k*(x^2+y^2));" "6#/-%\"u G6$%\"xG%\"yG*&%\"AG\"\"\"-%$expG6#*&%\"kGF+,&*$F'\"\"#F+*$F(F3F+F+F+ " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "pde := y*Dif f(u(x,y),x)=x*Diff(u(x,y),y);\npdsolve(pde,HINT=`*`):\nPDEtools[build] (%);\npdsolve(pde);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/*&%\"yG \"\"\"-%%DiffG6$-%\"uG6$%\"xGF'F/F(*&F/F(-F*6$F,F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG**%$_C1G\"\"\"-%$expG6#,$*&#F+\" \"#F+*&&%#_cG6#F+F+)F'F2F+F+F+F+%$_C2GF+-F-6#,$*&F1F+*&F4F+)F(F2F+F+F+ F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG-%$_F1G6#,&*$ )F'\"\"#\"\"\"F0*$)F(F/F0F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_______________________________ ______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 54 "Find s olutions for the partial differential equation " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Diff(u(x,y),x,y) = u(x,y);" "6#/-%%Di ffG6%-%\"uG6$%\"xG%\"yGF*F+-F(6$F*F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 29 "by separating the variables. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y) = A*exp(k*x+y/k);" "6#/-%\"uG6$% \"xG%\"yG*&%\"AG\"\"\"-%$expG6#,&*&%\"kGF+F'F+F+*&F(F+F1!\"\"F+F+" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "pde := Diff( u(x,y),x,y)=u(x,y);\npdsolve(pde,HINT = `*`):\ncombine(PDEtools[build] (%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/-%%DiffG6%-%\"uG6$%\" xG%\"yGF,F-F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG*( %$_C1G\"\"\"%$_C2GF+-%$expG6#,&*&&%#_cG6#F+F+F'F+F+*&F2!\"\"F(F+F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 37 "___________________________ __________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }} {PARA 0 "" 0 "" {TEXT -1 54 "Find solutions for the partial differenti al equation " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Di ff(u(x,y),x)" "6#-%%DiffG6$-%\"uG6$%\"xG%\"yGF)" }{XPPEDIT 18 0 "`` +` `" "6#,&%!G\"\"\"F$F%" }{XPPEDIT 18 0 "Diff(u(x,y),y) = (cos*x-sin*y)* u(x,y);" "6#/-%%DiffG6$-%\"uG6$%\"xG%\"yGF+*&,&*&%$cosG\"\"\"F*F0F0*&% $sinGF0F+F0!\"\"F0-F(6$F*F+F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "by separating the variables. " }}{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)=A*exp(sin*x+cos*y+k*(x-y))" "6#/ -%\"uG6$%\"xG%\"yG*&%\"AG\"\"\"-%$expG6#,(*&%$sinGF+F'F+F+*&%$cosGF+F( F+F+*&%\"kGF+,&F'F+F(!\"\"F+F+F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "pde := Diff(u(x,y),x)+Diff(u(x,y),y)=(cos(x)-sin( y))*u(x,y);\nsimplify(pdsolve(pde,HINT = `*`)):\nsimplify(PDEtools[bui ld](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/,&-%%DiffG6$-%\"uG 6$%\"xG%\"yGF-\"\"\"-F(6$F*F.F/*&,&-%$cosG6#F-F/-%$sinG6#F.!\"\"F/F*F/ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG*(%$_C1G\"\"\"% $_C2GF+-%$expG6#,**&&%#_cG6#F+F+F'F+!\"\"-%$sinG6#F'F+*&F2F+F(F+F+-%$c osG6#F(F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{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 53 "Find solutions for t he partial differential equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2" "6#*$%\"xG\"\"#" }{TEXT -1 1 " " }{XPPEDIT 18 0 " Diff(u(x,y),x,y)" "6#-%%DiffG6%-%\"uG6$%\"xG%\"yGF)F*" }{XPPEDIT 18 0 "``+3*y^2*u(x,y)=0" "6#/,&%!G\"\"\"*(\"\"$F&*$%\"yG\"\"#F&-%\"uG6$%\"x GF*F&F&\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "by separa ting the variables. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "u(x,y)=A*exp(-k/x-y^3/k)" "6#/-%\"uG6$%\"xG%\"yG*&%\"AG \"\"\"-%$expG6#,&*&%\"kGF+F'!\"\"F2*&F(\"\"$F1F2F2F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "pde := x^2*Diff(u(x,y),x,y)+3*y^2*u(x,y)=0;\npdsolve(pde,HINT=`* `):\ncombine(PDEtools[build](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %$pdeG,&*&)%\"xG\"\"#\"\"\"-%%DiffG6%-%\"uG6$F(%\"yGF(F1F*F**(\"\"$F*) F1F)F*F.F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG*(% $_C1G\"\"\"-%$expG6#,&*&&%#_cG6#F+F+F'!\"\"F4*&F(\"\"$F1F4F4F+%$_C2GF+ " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{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 5 "Q7 " }}{PARA 0 "" 0 "" {TEXT -1 53 "Find solutions for the partial \+ differential equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),x,y);" "6#-%%DiffG6%-%\"uG6$%\"xG%\"yGF)F*" } {XPPEDIT 18 0 "``-2*x;" "6#,&%!G\"\"\"*&\"\"#F%%\"xGF%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " Diff(u(x,y),y)=u(x,y)" "6#/-%%DiffG6$-%\"uG6 $%\"xG%\"yGF+-F(6$F*F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "by separating the variables. " }}{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)=A*exp(x^2+k*x+y/k)" "6#/-%\"uG6$%\"xG%\"y G*&%\"AG\"\"\"-%$expG6#,(*$F'\"\"#F+*&%\"kGF+F'F+F+*&F(F+F3!\"\"F+F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "pde := Diff(u(x,y),x,y)-2*x*Diff(u(x,y),y)=u(x, y);\npdsolve(pde,HINT=`*`):\ncombine(PDEtools[build](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/,&-%%DiffG6%-%\"uG6$%\"xG%\"yGF-F.\"\" \"*(\"\"#F/F-F/-F(6$F*F.F/!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"uG6$%\"xG%\"yG*(%$_C1G\"\"\"-%$expG6#,(*$)F'\"\"#F+F+*&&%#_cG6#F2! \"\"F'F+F+*&F4F+F(F+F+F+%$_C2GF+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_______________________________ ______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q8 " }}{PARA 0 "" 0 "" {TEXT -1 53 "Find s olutions for the partial differential equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(u(x,y),`$`(x,2));" "6#-%%DiffG6$-% \"uG6$%\"xG%\"yG-%\"$G6$F)\"\"#" }{XPPEDIT 18 0 "``-``;" "6#,&%!G\"\" \"F$!\"\"" }{XPPEDIT 18 0 " Diff(u(x,y),y)=u(x,y)" "6#/-%%DiffG6$-%\"u G6$%\"xG%\"yGF+-F(6$F*F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "by separating the variables. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "u(x,y)=A*exp(k*x+(k^2-1)*y)+B*exp(-k*x+(k^2-1)* y)" "6#/-%\"uG6$%\"xG%\"yG,&*&%\"AG\"\"\"-%$expG6#,&*&%\"kGF,F'F,F,*&, &*$F2\"\"#F,F,!\"\"F,F(F,F,F,F,*&%\"BGF,-F.6#,&*&F2F,F'F,F7*&,&*$F2F6F ,F,F7F,F(F,F,F,F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "u(x,y)=exp(-(k^2+1)* y)*(A*cos*k*x+B*sin*k*x)" "6#/-%\"uG6$%\"xG%\"yG*&-%$expG6#,$*&,&*$%\" kG\"\"#\"\"\"F3F3F3F(F3!\"\"F3,&**%\"AGF3%$cosGF3F1F3F'F3F3**%\"BGF3%$ sinGF3F1F3F'F3F3F3" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "pde := Diff(u(x,y),x$2)-Diff (u(x,y),y)=u(x,y);\npdsolve(pde,HINT=`*`):\ncombine(PDEtools[build](%) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/,&-%%DiffG6$-%\"uG6$%\"x G%\"yG-%\"$G6$F-\"\"#\"\"\"-F(6$F*F.!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG,&*(%$_C3G\"\"\"%$_C1GF,-%$expG6#,(* &&%#_cG6#F,#F,\"\"#F'F,F,*&F3F,F(F,F,F(!\"\"F,F,*(F+F,%$_C2GF,-F/6#,(F 2F9F8F,F(F9F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{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 "" }}{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 }