While trying to solve for the Laplacian in polar coordinates, one encounters the famous Euler-Cauchy differential equation.

or

How does one find a solution to this differential equation ? Well, most places that I have read simply dictate that the solutions to this differential equation as without explaining why only these two are the only solutions. The purpose of this post is to explain why!

We can clearly see that x = 0 is a singular point. Therefore a simple power series solution won’t work. Hence we use the frobenius method to get rid of the singularity i.e

Let’s now compute its derivatives wrt x.

Putting the values for and back into the differential equation, we get the following form.

Bringing the terms inside the summation, we obtain the following form:

Obviously, the coefficients of the summation cannot be zero and x=0 is a trivial solution. Therefore, we get the indicial equation.

or

(i) When n = 0, the indicial equation becomes

The values of the that solve for this equation are

(ii) When n = 1, the indicial equation becomes

The values of the that solve for this equation are

And in general:

,

Here is the crux of it all.

Lets now write down the solution for this differential equation in its glory:

where A and B are constants.

This is the general solution to the Euler-Cauchy differential equation. ðŸ˜€