Power-Series Solutions

2 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

The idea

When a useful elementary solution is unavailable, a power series can represent the unknown locally. At an ordinary point of a linear equation with analytic coefficients, substitution yields a recurrence determining coefficients from initial data.

Visual guide

VISUAL GUIDEA coefficient recurrence builds a local approximation

Exact cos x
1 − x²/2 + x⁴/24

For y″ + y = 0 with y(0) = 1 and y′(0) = 0, the recurrence produces 1 − x²/2! + x⁴/4! − ⋯. The fourth-degree truncation agrees well with cos x near zero but becomes inaccurate farther away.

Method and assumptions

Write $y=\sum_{n=0}^\infty a_nx^n$ . Differentiate term by term inside the convergence interval, shift indices to use the same power of $x$ , and equate coefficients. For a second-order equation, $a_0=y(0)$ and $a_1=y'(0)$ supply the two degrees of freedom.

Worked example: recovering trigonometric functions

For $y''+y=0$ , coefficient matching gives $(n+2)(n+1)a_{n+2}+a_n=0$ . With $a_0=1,a_1=0$ , the series is $1-x^2/2!+x^4/4!-\cdots=\cos x$ .

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

For y′=y and y=Σₙ₌₀^∞ aₙtⁿ, what coefficient relation follows?

aₙ₊₁=(n+1)aₙ

aₙ₊₁=0

(n+1)aₙ₊₁=aₙ

Hint 1 · Find a starting point

Differentiate the power series and align powers tⁿ.

Hint 2 · Take the next step

The derivative coefficient at tⁿ is (n+1)aₙ₊₁.

Show the reasoning

Answer: (n+1)aₙ₊₁=aₙ

Matching it to aₙ yields the recurrence, giving aₙ=a₀/n!.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

Worked example: a variable coefficient

For $y'=xy$ , matching powers gives $a_1=0$ and $(n+1)a_{n+1}=a_{n-1}$ for $n\ge1$ . With $a_0=1$ , the result is $1+x^2/2+x^4/8+x^6/48+\cdots=e^{x^2/2}$ . The recurrence generates coefficients without guessing the exponential.

Interpreting the result

A truncated series is an approximation and needs an error or convergence argument for its intended interval. Singular points may require a Frobenius series rather than ordinary nonnegative integer powers; that is a further topic beyond this introductory method.

Practice

Find $a_2$ for $y''+y=0$ when $a_0=3$ .
Which coefficients vanish when $a_1=0$ ?
What data determine $a_0,a_1$ ?

Show worked solutions

$a_2=-a_0/2=-3/2$ .
Every odd coefficient, by the two-step recurrence.
The initial displacement and slope at the expansion point.

Further study

MIT OpenCourseWare: Differential Equations provides a full university course with additional lectures and exercises.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

Optional marks, not a grade. Saved in this browser only. Open notebook →