# Recurrence relation of coeeficients of power series solution to DE

Say I have a DE

$$-\phi \left(\phi \left(\left(6975 \phi ^2-3704 \phi +160\right) \omega '(\phi )+\phi \left(\left(6975 \phi ^2-4688 \phi +266\right) \omega ''(\phi )+\phi \left(2 \left(675 \phi ^2-518 \phi +32\right) \omega ^{(3)}(\phi )-\phi \omega ^{(4)}(\phi )\right)\right)\right)\right) = 0$$

and I want a recurrence relation for coefficients of a power series solution, such that the constant term $$\omega(0)=1$$, how can I do that?

I will change symbols to $$t$$ and $$f$$ because they will look nicer here

t[f_] := x D[f,x]
R[x_] := -9 x^3 + 5 x^2 - 4 x
L[f_,x_] := R[x] Nest[t,f,4] + R[x]^2 Nest[t,f,3] + f
diffEqn = Expand[L[h[x],x]]

• Can you at least rewrite your ODE in Mathematica's format? – J. M.'s technical difficulties Feb 4 at 16:49
• @J.M.isinlimbo i have edited the post – Bernoulli Feb 4 at 16:53
• Have you seen this? The only change needed in your case would be to use SeriesCoefficient[] instead of Series[], so that you (hopefully) get a DifferenceRoot[] object. – J. M.'s technical difficulties Feb 4 at 16:57
• @J.M.isinlimbo it doesn't give me a recurrence relation. If I only have one initial condition f[0]==1, it would say "not a well-formed linear differential equation with initial conditions." – Bernoulli Feb 4 at 17:18