Explicit analytic solution of an 2D Poisson equation

I am not very familiar with analytic solution of PDEs. Here is the problem I don't know how to solve:

Let $\Omega$ be the unit square $(0,1)^2$, we consider the elliptic equation

$-div(k(x,y) \nabla u(x,y))=1$ in $\Omega$,
$u(x,y)=0$ on $\partial \Omega$,

where $k(x,y) = .2(2.5 + 1.5\sin (2\pi x))(2.5 + 1.5\cos (2\pi y)).$

Does we have an explicit analytic solution for this equation?

Thank you!

edited Jul 21 at 13:00

Andrew
1,013●1●4●9

asked Jul 21 at 10:16

amorphous
1

I know how to solve it numerically by using finite element method, but don't know how find the analytic one. I am not a math major, so please forgive me if this question is too basic. – amorphous Jul 21 at 12:01

Most PDEs do not have explicit solutions. But this problem was posed in a textbook? Since it has the square as region, it suggests to separate variables. Attempt a solution of the form $u(x,y) = f(x)g(y)$, determine ODEs for $f$ and $g$. – Gerald Edgar Jul 21 at 13:21

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