Solution
= Solution
Here the velocity contribution in <D'Alembert's formula> is
$$
\frac12\int_{x-t}^{x+t}e^s\,ds=\frac{e^{x+t}-e^{x-t}}2=e^x\sinh t.
$$
Thus, with arbitrary twice continuously differentiable $h$,
$$
\boxed{u(x,t)=\frac{h(x+t)+h(x-t)}2+e^x\sinh t.}
$$
For the added term, $u_{tt}=e^x\sinh t=u_{xx}$ and $u_t(x,0)=e^x\cosh0=e^x$. Its displacement at $t=0$ is zero, leaving $h$ free as required.