Find a power series solution to the differential equation at the point x0. (2+x2)y′′-xy′+4y=0

I'll assume you are looking for a series centered at , which is an ordinary point for the ODE. Substituting



into the ODE, we can rewrite it as









so the coefficients of the power series are defined by the recurrence



For even , i.e. , we have







and so on. The pattern in the denominator is pretty clear, but we can also find a compact form for the numerator. When , we can write it as







So in general we have



We can treat the odd-indexed terms similarly. For we have






and so on. Again, the pattern in the denominator is simple. For , we would get a numerator of






and in general we'd have



Thus the power series solution to this ODE is



Attached below is a plot of a numerical solution (blue) compared to the first 9 terms and first 21 terms of the series solution over the interval , assuming initial values of .

Hold on lemme do the work