Added Oct 8, 2019.
Exam problem. Math 5587, fall 2019. UMN
solve for \(u(x,t)\) the PDE \(u_t+\frac {1}{x^2+4} u_x= 0\) IC \(u(x,0)=e^{x^3+12 x}\)
Mathematica ✓
ClearAll["Global`*"]; pde = D[u[x,t], t] +1/(x^2+4)*D[u[x,t], x]== 0; ic = u[x,0]==Exp[x^3+12*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[{pde,ic}, u[x,t], {x,t}], 60*10]];
\[\left \{\left \{u(x,t)\to e^{-3 t+x^3+12 x}\right \}\right \}\]
Maple ✓
restart; pde := diff(u(x,t), t) +1/(x^2+4)*diff(u(x,t),x) =0; ic:=u(x,0)=exp(x^3+12*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde,ic],u(x,t))),output='realtime')); sol:=simplify(expand(sol));
\[u \left ( x,t \right ) ={{\rm e}^{{x}^{3}-3\,t+12\,x}}\]
Hand solution
Solve \[ u_{t}+\frac {1}{x^{2}+4}u_{x}=0 \] with initial conditions \(u\left ( x,0\right ) =e^{x^{3}+12x}\).
Solution
Let \(u=u\left ( x\left ( t\right ) ,t\right ) \). Then \begin {equation} \frac {du}{dt}=\frac {\partial u}{\partial x}\frac {dx}{dt}+\frac {\partial u}{\partial t}\tag {2} \end {equation} Comparing (1),(2) shows that \begin {align} \frac {du}{dt} & =0\tag {3}\\ \frac {dx}{dt} & =\frac {1}{x^{2}+4}\tag {4} \end {align}
Solving (3) gives\begin {align} u & =u\left ( x\left ( 0\right ) \right ) \nonumber \\ & =e^{x\left ( 0\right ) ^{3}+12x\left ( 0\right ) }\tag {5} \end {align}
We just need to find \(x\left ( 0\right ) \) to finish the solution. From (4)\begin {equation} \frac {x^{3}}{3}+4x=t+C\tag {6} \end {equation}
At \(t=0\)
\[ \frac {x\left ( 0\right ) ^{3}}{3}+4x\left ( 0\right ) =C \]
Hence (6) becomes
\begin {align*} \frac {x^{3}}{3}+4x & =t+\frac {x\left ( 0\right ) ^{3}}{3}+4x\left ( 0\right ) \\ x\left ( 0\right ) ^{3} & =3\left ( \frac {x^{3}}{3}+4x-t-4x\left ( 0\right ) \right ) \\ & =x^{3}+12x-3t-12x\left ( 0\right ) \end {align*}
Substituting this back into (5) gives\begin {align*} u\left ( x\left ( t\right ) ,t\right ) & =\exp \left ( x^{3}+12x-3t-12x\left ( 0\right ) +12x\left ( 0\right ) \right ) \\ & =e^{x^{3}+12x-3t} \end {align*}
The following is an animation of the solution
Source code used for the above
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