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Series equation to solve for coefficient A_k

Where m = ω = h = c1 = x0 = constants

I'm trying to solve a second order differential equation with non-constant coefficients and I using series to solve for it. I've plugged in the series solution and have simplified it down to the above equation. I have gotten everything to be represented in x^k, but I can't get the series in the second and third term to start from the same index without writing out some terms.

I want to represent everything in one series with x^k so that I can equate the coefficients equal to zero. Then I can find the reoccurance relation to find what A_k equals to for the solution to the differential equation I'm trying to solve.

Is it possible to solve for A_k?

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You can just start plugging in values of k to get a few relations. – sizz Aug 24 '12 at 19:47
This expression comes from what ODE? – doraemonpaul Dec 12 '12 at 18:28

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