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Consider the PDE given by $u_{tt}=c^2u_{xx}+f(x)$ representing a vibrating string with an external force acting on it which is independent of time but depends on position.

Show that the PDE has a time-independent solution $u=h(x)$ describing how $h$ is obtained from $f$.

Plugging $u=h(x)$ into the PDE gives \begin{align*} &0=c^2h''(x)+f(x) \\ & \implies h''(x)=-\frac{f(x)}{c^2} \end{align*}

What do I need to do from here?

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    If you don't know anything about $f$, eigenfunction expansion is most probably your best bet. – 2017-02-17
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    Do you have any properties on $f$ and the domain ? With sufficient conditions, you can define $h$ as a twice antiderivative of $-f/c^2$. – 2017-02-17
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    @Groovy. I'm not told anything about $f$. What conditions are necessary for me to define $h$ as you described above? – 2017-02-17
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    It would be sufficient that $f$ be continuous, or even piecewise continuous. Then you can just say that $$h(x) = \int_0^x\int_0^p -f(b)/c^2\,db\,dp$$ modifying the lower bounds to fit the domain, adding an affine function. – 2017-02-17

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