# 2-D Heat Equation IVP

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.

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.