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I am studying heat equations and I am trying to categorize the questions and their general solution as well as techniques to solve them. I reached to this excercise:
for half line $0<x<\infty$
$u_t-u_{xx}=0$
$u(x,0)=sin(2x)$
$u(0,t)=0$
I solved the problem to some point but I am stuck with this integral. I do not know how to solve this:
$\int_{-\infty}^{\infty} e^{\frac{-(x-y)^2}{4t}}sin(2y) \,dy$ I would appreciate any help with this

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Separation of variables might make you guess: $$ u(x,t)= e^{-4t}\sin(2x) $$

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