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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 23 "Problems of the day #15
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 1 "1" }{TEXT 278 61 ". Solve the
 following PDE and get Maple to plot the solution:" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/,&-%%diffG6$-%\"uG6$%\"xG%\"tGF,\"\"\"*&F,F--F&6$F(F
+F-F-\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG\"\"!*&\"
\"\"F*,&\"\"\"F,*$)F'\"\"#F*F,!\"\"" }}}{EXCHG {PARA 259 "" 0 "" 
{TEXT 257 8 "Method 1" }{TEXT -1 10 ": By hand." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 
22 "Step 1.  Solve the ODE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diff
G6$-%\"XG6#%\"tGF*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"XG6#\"\"!
%#xiG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "dsolve(\{diff(X(t)
,t)=t, X(0)=xi\}, X(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"XG6#%
\"tG,&*$)F'\"\"#\"\"\"#\"\"\"F+%#xiGF." }}}{EXCHG {PARA 261 "" 0 "" 
{TEXT -1 33 "Step 2.  Solve this equation for " }{XPPEDIT 18 0 "xi;" "
6#%#xiG" }{TEXT -1 26 " in terms of x=X(t) and t:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 22 "eqn := subs(X(t)=x,%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$eqnG/%\"xG,&*$)%\"tG\"\"#\"\"\"#\"\"\"F+%#xiGF." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(eqn,xi);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"\"*$)%\"tG\"\"#\"\"\"#!\"\"F)" }}}
{EXCHG {PARA 262 "" 0 "" {TEXT -1 35 "Step 3. The solution is u(x,t) =
 u(" }{XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 11 ",0), where " }
{XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 29 " is what you just solved f
or:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "u(x,t) := subs(xi=%,
 1/(1+xi^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"uG6$%\"xG%\"tG*&
\"\"\"F*,&\"\"\"F,*$),&F'F,*$)F(\"\"#F*#!\"\"F2F2F*F,!\"\"" }}}{EXCHG 
{PARA 257 "" 0 "" {TEXT 258 8 "Method 2" }{TEXT -1 14 ": Using Maple.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(PDEtools):" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 47 "eqn := diff(u(x,t),t) + t * diff(u(x,t),x) \+
= 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,&-%%diffG6$-%\"uG6$%
\"xG%\"tGF.\"\"\"*&F.F/-F(6$F*F-F/F/\"\"!" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "pdsolve (eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
-%\"uG6$%\"xG%\"tG-%$_F1G6#,&F'!\"#*$)F(\"\"#\"\"\"\"\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "u := unapply(subs(%, u(x,t)),x,t);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGR6$%\"xG%\"tG6\"6$%)operator
G%&arrowGF)-%$_F1G6#,&9$!\"#*$)9%\"\"#\"\"\"\"\"\"F)F)F)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "u(x,0);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$_F1G6#,$%\"xG!\"#" }}}{EXCHG {PARA 258 "" 0 "" 
{TEXT -1 61 "Therefore _F1(-2x) = 1/(1+x^2).  Thus _F1(x) = 1/(1+(x/2)
)^2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "_F1 := x -> 1/(1+(x
/2)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$_F1GR6#%\"xG6\"6$%)opera
torG%&arrowGF(*&\"\"\"F-,&\"\"\"F/*$)9$\"\"#F-#F/\"\"%!\"\"F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "u(x,t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&\"\"\"F$,&\"\"\"F&*$),&%\"xG!\"#*$)%\"tG\"\"#F$F&F/F$
#F&\"\"%!\"\"" }}}{EXCHG {PARA 263 "" 0 "" {TEXT -1 22 "Plotting the s
olution:" }}}{EXCHG {PARA 264 "" 0 "" {TEXT 259 8 "Method 1" }{TEXT 
-1 33 ": Using the solution found above:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 
"u(x,t) := 1/(1+1/4*(-2*x+t^2)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>-%\"uG6$%\"xG%\"tG*&\"\"\"F*,&\"\"\"F,*$),&F'!\"#*$)F(\"\"#F*F,F3F*#F
,\"\"%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot3d(u(x,t
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bfm7%$\"1*****>UotW%F-FKFbfm7%$\"1*****>U='GZF-FPFbfm7%$\"1*****>UoB2&
F-FUFbfmF(F(F(F(77F(F(F(F(F(F(F(F(F(7%$\"1+++6U$\\]%F-F.Fho7%$\"+6UotW
F]pF7F^p7%F[hmF<Fho7%$\"1+++6Uo)f%F-FAFho7%$\"1+++6U$\\v%F-FFFho7%$\"1
+++6Uot\\F-FKFho7%$\"1+++6U$\\D&F-FPFhoF(F(F(F(F(77F(F(F(F(F(F(F(F(F(7
%$\"1+++++DJ]F-F.F17%$\"*++++&!\")F7F87%F_imF<F1F(F(F(F(F(F(F(F(F(-%'C
URVESG6%76F4FjoFbrFitF`wFhyF_\\lFf^lF]alFeclF\\flFchlFjjlFa]mFh_mF_bmF
cdmFdfmF]hmFaim-%'COLOURG6&%$RGBGF7F7F7-%*THICKNESSG6#FL-%+AXESLABELSG
6%%\"xG%\"tGQ'u(x,t)6\"-%&STYLEG6#%-PATCHCONTOURG-%*AXESSTYLEG6#%&FRAM
EG-%%VIEWG6%;F5$\"\"&F7;F7Fd[n;F8Fccl" 3 354 354 354 4 0 1 0 2 1 0 3 
2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 
0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT 269 1 "2" }{TEXT 279 61 ". Solve the fol
lowing PDE and get Maple to plot the solution:" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,&-%%diffG6$-%\"uG6$%\"xG%\"tGF,\"\"\"*(F+F-F,F--F&6$F
(F+F-F-\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG\"\"!-%$
sinG6#F'" }}}{EXCHG {PARA 268 "" 0 "" {TEXT 270 8 "Method 1" }{TEXT 
-1 10 ": By hand." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart
;" }}}{EXCHG {PARA 269 "" 0 "" {TEXT -1 22 "Step 1.  Solve the ODE" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"XG6#%\"tGF**&F'\"\"\"F*
F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"XG6#\"\"!%#xiG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "dsolve(\{diff(X(t),t)=X(t)*t, X(0)=
xi\}, X(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"XG6#%\"tG*&%#xiG
\"\"\"-%$expG6#,$*$)F'\"\"#\"\"\"#F*F1F*" }}}{EXCHG {PARA 270 "" 0 "" 
{TEXT -1 33 "Step 2.  Solve this equation for " }{XPPEDIT 18 0 "xi;" "
6#%#xiG" }{TEXT -1 26 " in terms of x=X(t) and t:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 22 "eqn := subs(X(t)=x,%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$eqnG/%\"xG*&%#xiG\"\"\"-%$expG6#,$*$)%\"tG\"\"#\"\"
\"#F)F1F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(eqn,xi);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG\"\"\"-%$expG6#,$*$)%\"tG\"
\"#F%#\"\"\"F-!\"\"" }}}{EXCHG {PARA 271 "" 0 "" {TEXT -1 35 "Step 3. \+
The solution is u(x,t) = u(" }{XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 
11 ",0), where " }{XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 29 " is what
 you just solved for:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "u(
x,t) := subs(xi=%, sin(xi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"u
G6$%\"xG%\"tG-%$sinG6#*&F'\"\"\"-%$expG6#,$*$)F(\"\"#F-#\"\"\"F4!\"\"
" }}}{EXCHG {PARA 266 "" 0 "" {TEXT 271 8 "Method 2" }{TEXT -1 14 ": U
sing Maple." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(PDEtools):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "eqn := diff(u(x,t),t) + x * t * dif
f(u(x,t),x) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,&-%%diffG
6$-%\"uG6$%\"xG%\"tGF.\"\"\"*(F-F/F.F/-F(6$F*F-F/F/\"\"!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "pdsolve (eqn);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"uG6$%\"xG%\"tG-%$_F1G6#,&-%#lnG6#F'!\"#*$)F(\"\"#
\"\"\"\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "u := unappl
y(subs(%, u(x,t)),x,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGR6$%
\"xG%\"tG6\"6$%)operatorG%&arrowGF)-%$_F1G6#,&-%#lnG6#9$!\"#*$)9%\"\"#
\"\"\"\"\"\"F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "u(x,0)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$_F1G6#,$-%#lnG6#%\"xG!\"#" }}}
{EXCHG {PARA 267 "" 0 "" {TEXT -1 64 "Therefore _F1(-2 ln(x)) = sin(x)
.  Thus _F1(x) = sin(exp(-x/2))." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "_F1 := x -> sin(exp(-x/2));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$_F1GR6#%\"xG6\"6$%)operatorG%&arrowGF(-%$sinG6#-%$ex
pG6#,$9$#!\"\"\"\"#F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "simplify(u(x,t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#*&%
\"xG\"\"\"-%$expG6#,$*$)%\"tG\"\"#\"\"\"#!\"\"F0F(" }}}{EXCHG {PARA 
272 "" 0 "" {TEXT -1 22 "Plotting the solution:" }}}{EXCHG {PARA 273 "
" 0 "" {TEXT 272 8 "Method 1" }{TEXT -1 33 ": Using the solution found
 above:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "u(x,t) := sin(x*exp(-1/2*t^2
));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"uG6$%\"xG%\"tG-%$sinG6#*&F
'\"\"\"-%$expG6#,$*$)F(\"\"#\"\"\"#!\"\"F4F-" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 30 "plot3d(u(x,t),x=-5..5,t=0..5);" }}{PARA 13 "" 1 
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{TEXT 313 1 "3" }{TEXT 314 61 ". Solve the following PDE and get Maple
 to plot the solution:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%%diffG6
$-%\"uG6$%\"xG%\"tGF,\"\"\"*&F(F--F&6$F(F+F-F-\"\"!" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%\"uG6$%\"xG\"\"!F'" }}}{EXCHG {PARA 287 "" 0 "" 
{TEXT 309 8 "Method 1" }{TEXT -1 10 ": By hand." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 293 "" 0 "" {TEXT -1 
49 "This is Burgers' equation with initial condition:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "u0 := x -> x;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#u0GR6#%\"xG6\"6$%)operatorG%&arrowGF(9$F(F(F(" }}}
{EXCHG {PARA 288 "" 0 "" {TEXT -1 42 "Step 1.  Solve the following equ
ation for " }{XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 1 ":" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn := x = u0(xi) * t + xi;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/%\"xG,&*&%#xiG\"\"\"%\"tGF*F*F
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{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG\"\"\",&%\"tG\"\"\"F(F(!\"\"" }
}}{EXCHG {PARA 289 "" 0 "" {TEXT -1 36 "Step 2. The solution is u(x,t)
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or:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "u(x,t) := subs(xi=%,
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"with(PDEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "eqn := \+
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{XPPMATH 20 "6#>%$eqnG/,&-%%diffG6$-%\"uG6$%\"xG%\"tGF.\"\"\"*&F*F/-F(
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G6#,(*&-%$_F1G6#%#_ZG\"\"\"F1F2F2*&F(F2F1\"\"\"!\"\"F'F2" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "u := unapply(subs(%, u(x,t)),x,t);
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G%&arrowGF)-%'RootOfG6#,(*&-%$_F1G6#%#_ZG\"\"\"F5F6F6*&9%F6F5\"\"\"!\"
\"9$F6F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "u(x,0);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RootOfG6#,&*&-%$_F1G6#%#_ZG\"\"\"F+
F,F,%\"xGF," }}}{EXCHG {PARA 286 "" 0 "" {TEXT -1 31 "Here Z = u0(x), \+
so _F1(x) = -1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "_F1 := x
 -> -1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$_F1G!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "u(x,t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&%\"xG\"\"\",&\"\"\"F'%\"tGF'!\"\"" }}}{EXCHG {PARA 
290 "" 0 "" {TEXT -1 22 "Plotting the solution:" }}}{EXCHG {PARA 291 "
" 0 "" {TEXT 311 8 "Method 1" }{TEXT -1 33 ": Using the solution found
 above:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
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}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT 324 1 "4" }{TEXT 329 61 ". Solve the following PDE and get Map
le to plot the solution:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%%diff
G6$-%\"uG6$%\"xG%\"tGF,\"\"\"*&F(F--F&6$F(F+F-F-\"\"!" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%\"uG6$%\"xG\"\"!-%*PIECEWISEG6$7$F(2F'F(7$F'%*o
therwiseG" }}}{EXCHG {PARA 296 "" 0 "" {TEXT 319 8 "Method 1" }{TEXT 
-1 10 ": By hand." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart
;" }}}{EXCHG {PARA 302 "" 0 "" {TEXT -1 49 "This is Burgers' equation \+
with initial condition:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "
u0 := x -> piecewise(x < 0, 0, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%#u0GR6#%\"xG6\"6$%)operatorG%&arrowGF(-%*piecewiseG6%29$\"\"!F1F0F(F
(F(" }}}{EXCHG {PARA 297 "" 0 "" {TEXT -1 42 "Step 1.  Solve the follo
wing equation for " }{XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 1 ":" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn := x = u0(xi) * t + xi;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/%\"xG,&*&-%*PIECEWISEG6$7$
\"\"!2%#xiGF-7$F/%*otherwiseG\"\"\"%\"tGF2F2F/F2" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 14 "solve(eqn,xi);" }}}{EXCHG {PARA 303 "" 0 "" 
{TEXT -1 71 "Maple fails to solve the equation, so do it by hand (vali
d for t > -1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "xi = piec
ewise(x<0,x,x/(t+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#xiG-%*PIEC
EWISEG6$7$%\"xG2F)\"\"!7$*&F)\"\"\",&%\"tG\"\"\"F1F1!\"\"%*otherwiseG
" }}}{EXCHG {PARA 298 "" 0 "" {TEXT -1 36 "Step 2. The solution is u(x
,t) = u0(" }{XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 9 "), where " }
{XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 29 " is what you just solved f
or:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "u(x,t) := subs(%, u0
(xi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"uG6$%\"xG%\"tG-%*PIECEW
ISEG6$7$\"\"!2-F*6$7$F'2F'F-7$*&F'\"\"\",&F(\"\"\"F7F7!\"\"%*otherwise
GF-7$F/F9" }}}{EXCHG {PARA 304 "" 0 "" {TEXT -1 77 "Maple doesn't know
 how to simplify this, so do it by hand (valid for t > -1):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "u(x,t) := piecewise (x < 0, \+
0, x/(t+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"uG6$%\"xG%\"tG-%*
PIECEWISEG6$7$\"\"!2F'F-7$*&F'\"\"\",&F(\"\"\"F3F3!\"\"%*otherwiseG" }
}}{EXCHG {PARA 294 "" 0 "" {TEXT 320 8 "Method 2" }{TEXT -1 14 ": Usin
g Maple." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(PDEtools):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "eqn := diff(u(x,t),t) + u(x,t) * di
ff(u(x,t),x) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,&-%%diff
G6$-%\"uG6$%\"xG%\"tGF.\"\"\"*&F*F/-F(6$F*F-F/F/\"\"!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "pdsolve (eqn);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"uG6$%\"xG%\"tG-%'RootOfG6#,(*&-%$_F1G6#%#_ZG\"\"\"
F1F2F2*&F(F2F1\"\"\"!\"\"F'F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 34 "u := unapply(subs(%, u(x,t)),x,t);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"uGR6$%\"xG%\"tG6\"6$%)operatorG%&arrowGF)-%'RootOfG6#,(*&-%$
_F1G6#%#_ZG\"\"\"F5F6F6*&9%F6F5\"\"\"!\"\"9$F6F)F)F)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 7 "u(x,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%'RootOfG6#,&*&-%$_F1G6#%#_ZG\"\"\"F+F,F,%\"xGF," }}}{EXCHG {PARA 
295 "" 0 "" {TEXT -1 85 "Here Z = u0(x), and so _F1(x) = -1 if x > 0. \+
However, can't solve for _F1 when x < 0." }}}{EXCHG {PARA 305 "" 0 "" 
{TEXT -1 57 "The following solution is therefore only valid for x > 0:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "_F1 := x -> -1;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$_F1G!\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 7 "u(x,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"x
G\"\"\",&\"\"\"F'%\"tGF'!\"\"" }}}{EXCHG {PARA 299 "" 0 "" {TEXT -1 
22 "Plotting the solution:" }}}{EXCHG {PARA 300 "" 0 "" {TEXT 321 8 "M
ethod 1" }{TEXT -1 33 ": Using the solution found above:" }}}{EXCHG 
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