2.2.4 \(u_{tt} - 2 u_{xt} - 3 u_{xx} = 0\) with \(u(0,x)=x^2, u_t(x,0)=e^x\)

problem number 82

Added Oct 6, 2019.

Problem 2.4.19 Peter Olver, Into to Partial differential equations 4th edition

Solve \(u_{tt} - 2 u_{xt} - 3 u_{xx} = 0\) with \(u(0,x)=x^2, u_t(x,0)=e^x\)

Mathematica ✓

ClearAll["Global`*"]; 
pde = D[u[x, t], {t,2}] - 2*D[D[u[x,t],x],t] -  3*D[u[x, t], {x,2}] == 0; 
ic  = {u[x,0]==x^2, Derivative[0,1][u][x,0]==Exp[x]}; 
sol = AbsoluteTiming[TimeConstrained[DSolve[{pde,ic}, u[x, t], {x, t}], 60*10]];
 

\[\left \{\left \{u(x,t)\to 3 t^2-\frac {e^{x-t}}{4}+\frac {1}{4} e^{3 t+x}+x^2\right \}\right \}\]

Maple ✓

restart; 
pde := diff(u(x,t),t$2)-2*diff(diff(u(x,t),x),t) - 3 * diff(u(x,t),t$2)=0; 
ic  := u(x,0)=x^2, D[2](u)(x,0)=exp(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde,ic],u(x,t))),output='realtime'));
 

\[u \left ( x,t \right ) ={{\rm e}^{x}}-{{\rm e}^{-t+x}}+{x}^{2}\]