Series Solutions to Differential Equations

Calculus 2 · Axiom Academy

LESSON Series Solutions to Differential Equations Solving y' = y with a power series — and watching the exponential function build itself, one coefficient at a time. Consider the simplest differential equation, with an initial condition: We secretly know the answer is y = e^x — but let's pretend we don't, and rediscover it. Instead of guessing a closed form, we assume the solution can be written as a power series with unknown coefficients : Differentiating a power series is easy: each term x^n becomes . The power drops to the front as a coefficient, and the exponent decreases by one. Watch what happens to each term — and notice the constant term a_0 vanishes. differentiate term by term — the index shifts The equation says y' = y , so the two series must be equal . Line them up and compare them power by power. Matching the coefficient of x^k on each side gives one equation per power. Solving each for the next coefficient produces a recurrence : The initial condition y(0) = 1 pins down the first coefficient: setting x = 0 leaves only a_0 , so a_0 = 1 . Now the recurrence cascades — each new coefficient divides the last by the next integer. Watch the denominators stack up. Each denominator is a factorial: for all . Substitute the coefficients back into the series. The partial sums — 1 , then 1 + x , then , and so on — each hug the true exponential curve over a wider and wider range. In the limit, they are e^x .

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