Past exam of the year 1 of the computer science course of the University of Oxford / 2005 / Calculus, linear algebra, discrete mathematics, logic and proof, and models of computation / 4 / b / iii / Solution

Here the velocity contribution in D'Alembert's formula isThus, with arbitrary twice continuously differentiable $h$,For the added term, $u_{tt}=e^x\sinh t=u_{xx}$ and $u_t(x,0)=e^x\cosh 0=e^x$. Its displacement at $t=0$ is zero, leaving $h$ free as required.