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
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?