Separation of Variables (Partial Differential Equation)

Question

Does Separation of Variables work for the following PDE ?

$$\nabla^2 W(x,y) \pm \alpha W(x,y) = \beta,$$

where $\alpha$ and $\beta$ are constants.

Answer 1

Sure. First solve the homogeneous case, i.e. $\beta = 0$. Then, separating in the usual way we guess solutions of the form $W(x,y)=X(x)Y(y)$, yielding

$\frac{1}{X(x)}\frac{d^2X}{dx^2} \pm \alpha = -\frac{1}{Y(y)} \frac{d^2Y}{dy^2} = \lambda^2$

implying that

$\frac{d^2X}{dx^2}=X(x)(\lambda^2\mp \alpha) = X(x)k^2$

where $k^2 = \lambda^2\mp \alpha$. We also have

$\frac{d^2Y}{dy^2} +\lambda^2 Y(y)=0$.

I assume you can solve both of these, so I will move on.

Now you have $W(x,y)=X(x)Y(y)$ which have just been solved for in the above ODEs. Then, you will write $\beta$ in terms of a Fourier series (perhaps Sine or Cosine depending on your boundary conditions, or perhaps a mixture if you need it to be general). Plug your $W(x,y)$ series solution into the PDE. Then, we can bring the Fourier series for $\beta$ over to the other side and match term by term in the series. Now you have an equation (possible a DE) for your constant terms. Then, you have a general solution to the inhomogenous PDE.