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**flyingsquirrel** Hello,

Using partial fractions decomposition, $\displaystyle u(x,t)=\frac{1}{2a}\left( \frac{1}{at-x}+\frac{1}{at+x}\right)$. Now differentiate $\displaystyle u$ twice with respect to $\displaystyle t$, differentiate $\displaystyle u$ twice with respect to $\displaystyle x$ and you're done. If you keep the derivatives under the form $\displaystyle \text{constant}\times \left( \text{a fraction}+\text{another fraction}\right)$ it should be easier than the methods you've tried ; you simply need to know that the derivative of $\displaystyle \frac{1}{f}$ is $\displaystyle -\frac{f'}{f^2}$.