For the differential equation 3x^2y''+2xy'+x^2y=0 show that the point x = 0 is a regular singular point (either by using the limit definition or by computing the associated euler equation). compute the recursion formula for the series solution corresponding to the larger root of the indicial equation. with a0 = 1, compute the first three nonzero terms of the series.

Given an ODE of the form



a regular singular point is one such that or diverge as , but the limits of and as exist.

We have for ,



and as , we have and , so indeed is a regular singular point.

We then look for a series solution about the regular singular point of the form



Substituting into the ODE gives







From this we find the indicial equation to be



Taking , and in the term above we find . So we have



Since , all coefficients with an odd index will also vanish.

So the first three terms of the series expansion of this solution are



with , , and .

We need the pictures of the graphs