Axisymmetric Poisson equation

Main Question or Discussion Point

For a magnetostatics problem I seek the solution to the following equation

$$\frac{1}{x}\frac{d}{dx} \left( x \frac{dy(x)}{dx} \right) = -C^2 y(x)$$

(C a real constant) or equivalently

$$x \frac{d^2 y(x)}{dx^2} + \frac{dy(x)}{dx} + C^2 x y(x)=0$$

It seems so simple, but finding a particular solution beats me...is this solvable?

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Mute
Homework Helper
If you rescale variables to get rid of the C^2 it looks like you could get it into the form of the differential equation for a zeroth order Bessel function. The general equation for a Bessel function is:

$$x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = 0$$

So with alpha = 0, you could divide out an x (or equivalently mutliply your equation by one) and it matches your equation - you just need to scale out the constant. i.e., somehow you want to scale that last term such that $C^2xy \rightarrow xy$ with the other terms remaining unchanged.

That's great! Thank you very much...works like a charm!