Series Method at Ordinary Points

Differential Equations · Axiom Academy

LESSON Series Method at Ordinary Points Assume , substitute into the equation, and let the algebra hand you a recurrence relation for every coefficient. Write the differential equation in standard form, with the highest derivative isolated: 2. Assume a Power Series Solution Near the ordinary point x_0 = 0 , assume the solution has the form: 3. Substitute Into the Equation Substitute the series and its derivatives into the equation. Take a concrete case that has no elementary closed form: Substituting the series for y and y'' turns the equation into a sum of power towers, one power of x per rung — but the towers don't yet line up: 4. Shift Indices to Align the Powers To combine sums, relabel the summation index so every series carries the same power of x , starting at the same value . 5. Match Coefficients — the Recurrence Relation With every sum aligned to the same power x^m , combine them into one series. For this to vanish for every x , every coefficient must independently equal zero: the lone m=0 term, handled separately the recurrence relation, valid for Setting generates one solution; setting generates the other. Both come from the SAME recurrence: You've walked the full power-series machine: assume the series, substitute, shift indices to align powers, match coefficients, and let the recurrence generate two independent solutions. Scroll up to revisit any step.

This is the written version of the interactive lesson above. See the full Differential Equations course.