Joseph C. answered • 08/10/20

Teacher | Researcher | Programmer

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_{n}x^{n}

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_{n}x^{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_{n}x^{n-1}

y''(x) = ∑_{n=2} n (n-1) a_{n}x^{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!