2.10.3 \(u_{tt} - u_{xx} - \frac{2}{x} u_x = 0\) with \(u(x,0)=0,u_t(x,0)=g(x)\)

problem number 105

Added Oct 6, 2019

Problem 2.4.18 from Peter Olver, Intoduction to Partial differential equations, 4th edition.

Solve for \(u(x,t)\) \[ u_{tt} - u_{xx} - \frac{2}{x} u_x = 0 \]

With \(u(x,0)=0,u_t(x,0)=g(x)\). Note, in the book, it says to assume \(g(x)\) is even function. In the code below, this assumption is not used. When I find the correct way to implement this assumption in CAS, will have to re-run these.

Mathematica

Failed

Maple

\[u \left ( x,t \right ) =\sum _{n=0}^{\infty }{\frac{{t}^{2\,n+1} \left ( U\mapsto 0^{ \left ( n \right ) } \right ) \left ( g \left ( x \right ) \right ) }{ \left ( 2\,n+1 \right ) !}}\]