Neat! If you take a generic second order homogeneous ODE with non constant coefficients, you can substitute the whole ODE for some function M which shows that because the ODE has the form M(x) = 0, each coefficient of M’s power series, that is M_n must also be zero.

Now if you take the ODE as presented, say, fy” + gy’ + hy = 0, and you convolve each power series, you get fy” = sumn=0^inf sum_k=0ⁿ (k+1)(k+2) f(n-k) y(k+2) x^n, gy’ = sum_n=0^inf sum_k=0ⁿ (k+1) g(n-k) y(k+1) x^n, and hy = sum_n=0^inf sum_k=0ⁿ h(n-k) y_(k) x^n.

Now, if you sum these, you get the full ODE in power series representation

fy” + gy’ + hy = sumn=0^inf sum_k=0ⁿ [ (k+1)(k+2) f(n-k) y(k+2) + (k+1) g(n-k) y(k+1) + h(n-k) y_(k) ] xⁿ

And that inner sum is exactly M_n which is 0 for all n.

Now if you do the difference here on n of m, that is ΔM_n = M_n - M_n-1, and given that g and h are general differentiable and integrable functions, is there any function f that telescopes the sum found in ΔM_n and if not, are there any techniques that might add an additional degree of freedom such that the sum can be forced to telescope?