Power Series & Recurrence Relation Deriver

Reverse Engineer ODE from Target Constant

Target constant K

Derivation engine

Target x-value x*

Initial value a_0

ODE mode

Common x-power m

Gamma Lift

alpha-shifts (comma list)

beta-shifts (comma list)

Gamma lift scale c

Differential Equation

Use y' for y′ and y for y. Coefficients can be constants or monomials in x.
Examples: y' + y = 0 | xy' + 10y = 0 | x^2y' - 3xy = 0

Step 1 — Assume a Power Series Solution

We look for a solution of the form of a power series centered at \(x = 0\):

Step 2 — Differentiate Term by Term

Differentiating the series:

Step 3 — Substitute into the ODE

Step 4 — Re-index to Align Powers of x

Step 5 — Set Each Coefficient Equal to Zero

Since the series equals zero for all \(x\), every coefficient of \(x^n\) must vanish:

Step 6 — Recurrence Relation

Step 7 - Special Function Forms

Step 8 - Final Series Representation

Step 9 - Target Constant Closure

Power Series Recurrence Deriver