# Radius of convergence of a power series solution to a differential equation

If I had a power series given by $$\sum_{n=0}^{\infty}a_{n}x^{n}$$ and it was the solution to a differential equation, how would I go about finding the radius of convergence of said power series? TIA

• The same way you would find the radius of convergence of any power series. Apr 24 '20 at 11:33

$$y^{(n)} + p_{n-1}(x)y^{(n-1)} + \cdot +p_0(x)y = g(x)$$,
then you can find all the singularities in all the coefficient functions $$p_k(t)$$. The distance from $$0$$ to the nearest singularity is the radius of convergence.
Notes: If you center the series at $$x_0$$, then it's the distance from $$x_0$$ to the nearest singularity.
You have to account for the complex singularities. If one of your coefficients is $$1/(1+x^2)$$ then the radius of convergence is no more than the distance from $$0$$ to $$i$$.