Y"-3xy=0 given ordinary point x=0. power series

This generates the recurrence relation



Because you have



it follows that for all .

For , you have






so that, in general, for , , you have



Now, for , you have






and so on, with a general pattern for , , of



Putting everything together, we arrive at the solution




To show this solution is sufficient, I've attached is a plot of the solution taking and , with . (I was hoping to be able to attach an animation that shows the series solution (orange) converging rapidly to the exact solution (blue), but no such luck.)