Question

x^2y'' + xy' + (3+x)y=0
What method do I use to solve the ODE?

Answer 1

I can get you started. I think a power series solution would work, but I was checking through my books to see if this equation was an example of something else that's been solved exactly - I can't find such an equation.

The problem with the series solution is that, because the coefficient of y'' goes to zero as x goes to zero, you're going to have a singularity when you divide the equation through by x^2 (this method is actually called the Frobenius Method, if you want to look it up). You need to check if the singular point is regular or not. If it is, you can use the method.

I wish I could actually do the problem for you, but I'm running late.

I might check back later.

Hope this is enough to get you started.