TITLE: How to solve the Cauchy problem?
QUESTION [0 upvotes]: I have a task that looks like
$$ \frac{\partial u}{\partial t} = 2\frac{\partial^2 u}{\partial x^2} + \frac{\partial u}{\partial x} + u + 3e^{t},\, u\left(x, 0\right)=\sin{x}$$
Please help, thanks :)

REPLY [1 votes]: Some details:
$$\mathcal{L}^{-1}\left(\ \frac{3}{(s-1)^2}\right)=3te^t$$
$$\mathcal{L}^{-1}\left(\ \frac{1}{s^2+2s+2}\right)=e^{-t}\sin t$$
$$\mathcal{L}^{-1}\left(\ \frac{s}{s^2+2s+2}\right)=e^{-t}(\cos t-\sin t)$$
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