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Find the solution to the two-dimensional heat equation, $u_t = u_{xx} + u_{yy}$ in the $x$-$y$ plane (that is, $−∞ < x < ∞$, $−∞ < y < ∞$) with initial data $u(x,y,0) = xe^{-y}$

I'm only aware of the solution to the IVP of the heat equation in 1-D, that is $$u(x,t) = \frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{\infty} e^{-y^2/4t}u_0(x-y)dy$$

But I don't know how to go about this in two dimensions.

I also thought about doing Separation of Variables but without Boundary conditions, I don't know how to proceed.

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By inspection only, one see that a solution is : $$u(x,y,t)=xe^{-y+t} \tag 1$$ Proof :

$u_{xx}=0$

$u_{yy}=xe^{-y+t}$

$u_t=xe^{-y+t}$ $$u_{xx}+u_{yy}=0+xe^{-y+t}=xe^{-y+t}=u_t$$ Thus Eq.$(1)$ is solution of the PDE $u_t=u_{xx}+u_{yy}$ .

$$u(x,y,0)=xe^{-y+0}=xe^{-y}$$

Thus Eq.$(1)$ satisfies the initial condition.

Comment: A so elementary solution draw to think that probably there is something missing in the wording of the question.

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