Series solution of a second order differential equation with non-constant coefficients:
I can find the solution to an equation of the form (d^2y/dx^2) + y = 0 but what if the zero was replaced with another value, say x^2? Would it be brought over to make the equation equal to zero or would it be dealt with during the recurrence relation?

Question

Series solution of a second order differential equation with non-constant coefficients:
I can find the solution to an equation of the form (d^2y/dx^2) + y = 0 but what if the zero was replaced with another value, say x^2? Would it be brought over to make the equation equal to zero or would it be dealt with during the recurrence relation?

anonymous · Answer

What are you trying to solve exactly?

anonymous · Answer

I've come across several examples where the question asks: Determine a series solution for the following differential equation about x=x(subscript zero). One example is (d^2y/dx^2) + y = 0. I have no problem obtaining a solution for said examples. I'm curious as to how the process changes when the given equation does not equal zero but something else, say x squared.

anonymous · Answer

I believe when you solve for the recurrence the sums will pick out one value (namely n = 2 in a sum of x^n).

anonymous · Answer

I was thinking of doing that. Is that all that needs to be done? So what if the question was: Determine a series solution for the following differential equation about x=x0: $$(x^2-1)*(d^2y/dx^2) -8x(dy/dx) + 20y = 6x^2$$

anonymous · Answer

I get $$y=a0(1+10x^2+5x^4) + a1(x+2x^3+x^5/5) - x^4/2$$Is that a valid answer?