Can the Axisymmetric Poisson Equation for Magnetostatics be Solved?

[Wiemster]

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?

 

[Mute]

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.

 

[Wiemster]

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