Someone please help with this question!

Question

Use the method of Power Series to solve the following nonlinear ODE:

Please only use Power series method.

Answer 1

given that the problem requires a lot of detail. i will only be able to give you some hints of what to do

Power Series: y(x) = ∑_n=0 a_nxⁿ

This approach requires you to assume that y(x) has a power series form. Now you need to find the relevant derivatives y'(x) and y''(x)

Note on derivatives

y'(x) = ∑_n=? n a_nx^n-1

Note that if the index were zero the first term would be 1/x. So that would be a problem. Therefore n=1 is the first index. A similar thing happens for the second derivative

y'(x) = ∑_n=1 n a_nx^n-1

y''(x) = ∑_n=2 n (n-1) a_nx^n-2

Now the goal is to replace this derivatives on the differential equation and find conditions on the coefficients. Note that there will be 2 coefficients that you need to provide as boundary conditions since is a second order differential equation.

Hope this helps!