Assume that the differential equation

has a power-series solution and find a formula for the coefficient .

First, we have

So, from the given differential equation we have

Since each coefficient of must equal 0 for this equation to hold we have

By induction we then have

The coefficients and are arbitrary and we denote them by and respectively. Then we have

There are some mistakes, but the general idea is correct

The odd coefficient has wrong sign and wrong index. The resulting y missing the constant term.

How did you find a2n and a2n+1 by induction?