Solving a 2nd order ODE with variable coefficients

[AdrianZ]

Homework Statement

the problem is to solve this differential equation:

x^2 y'' + xy' + (4x^2 - 1)y = 0

 

[HallsofIvy]

I would recommend Frobenius' method, writing a solution as
$$y(x)= \sum_{n=0}^\infty a_nx^{n+c}$$
Find y'' and y' for that, put into the equation. Choose c (it is not necessarily a positive integer) such that the n= 0 coefficient is non-zero.

 

[Dickfore]

The solution is expressible in terms of Bessel functions, but you need to make the substitution:
$$t = a x$$
for the independent variable with a suitable choice of $a$.

 

[AdrianZ]

Actually this is a question for extra points. We haven't reached to Bessel function or Frobenius' method but the professor told us that if some of us have already passed discrete mathematics and known about generating functions then with doing some research on it we might be able to solve it although it's not an easy equation as he said. I tried to take the series that HallsofIvy suggested but didn't succeed to solve the problem.+ How can I find the value of c?

 

[Dickfore]

Try reading through this:

http://en.wikipedia.org/wiki/Frobenius_method