Find the solution u(x, y) of laplace's equation in the rectangle 0 < x < a, 0 < y < b, that satisfies the boundary conditionsu(0, y) = 0, u(a, y) = 0, 0< y< b, u(x, 0) = 0, u(x, b) = g(x), 0 ≤ x ≤ a.

Step-by-step explanation:

The Laplace's equation in rectangular coordinates is:

For the solution, we assume that u can be write as

Replace

Divide

The only way that this can be true, is that every term is the same constant, so we say that

With a constant.

Now, we solve every part. For

The solution is:

For

Is the differential equation for a harmonic oscillator, so the solution is

Now, we evaluate the boundary conditions:

The other

For :

So,

The other condition:

So,