Find the power series in x?

Question

Find the power series in x for the general solution of (1+2x^2)y"+7xy'+2y=0.

 

 

So I got up to a_n+2=-a_n(2n+1)/(n+1). How do I find the right answer?

 

 

Answer 1

So far, you have a recursion relationship.  This occurs when the y'' term has a second degree polynomial as a coefficient, the y' term has a first degree polynomial, and the y term has a zero degree polynomial.  Without checking your solution, I will assume your recursion relationship is correct.  Since the original differential equation is of second order, it should have two linearly independent solutions.  I would say that the terms with odd numbered powers add up to one solution while the terms with even numbered powers add up to a second solution, linearly independent of the first solution.  Start the odd series with the term a₁x¹ and start the even series with the term a₀x⁰, using the recursion relation to express the a_n coefficients.  The odd solution will be in terms of an unknown a₁ while the even solution will be in terms of an unknown a₀.  You will need boundary conditions to obtain a₁ and a₀.

 

I hope this helps.