Series Solution Builder

Differential Equations · Axiom Academy

Build a power series solution term by term — watch a differential equation collapse into a rule for its own coefficients. An infinite polynomial as the unknown Some differential equations have no solution built from familiar functions. So instead of guessing a formula, assume the solution IS a power series — — with every coefficient a_n unknown, then let the equation itself tell you what those coefficients have to be. Watch the series and its derivative get substituted into a simple equation, y'-y=0 . Every power of x must vanish on its own, so like terms slide together and collapse into a single rule — a recurrence relation — that generates every coefficient from the one before it. Every column collapses the same way — that's what makes it a general rule, not a one-off calculation. Turn the recurrence into the actual solution The recurrence you just watched appear is . Pick a starting value a_0 and how many terms to build, and every coefficient below is computed live from that one rule — then the partial sum is plotted against the curve it's aiming for. Slide N up: the chunky partial-sum curve tightens onto a_0e^x — the recurrence really does rebuild the exponential, one coefficient at a time. The SAME method, a different equation Switch the differential equation and watch everything downstream recompute — the recurrence formula, the coefficients, and the curve the partial sum is chasing. The method doesn't change; only what it produces does.

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