The power series method · Differential Equations

Intuition

Substitute the series into the equation, shift the indices so that every sum is written in the same power of x, and collect: the result is one power series that must be zero, so every one of its coefficients must be zero. Each of those conditions is an equation linking a few coefficients of the solution. Worked through for an equation whose answer is already known, the method finds that answer, which is the best evidence that it works.

Balancing a chemical equation atom by atom. Everything on one side must be matched on the other, element by element; here, power by power.

plane diagram: the-series-finds-the-cosine

$y''+y=0$ with $y(0)=1$ , $y'(0)=0$ . The recurrence $a_{n+2}=-\tfrac{a_{n}}{(n+2)(n+1)}$ gives $1-\tfrac{x^{2}}{2!}+\tfrac{x^{4}}{4!}-\cdots$ , and its partial sums of degree $2$ and $6$ close in on $\cos x$ , dashed. All three are even, so the right half says everything: the method finds the answer chapter four gave.

Substitute, shift, match

To solve $y''+p(x)y'+q(x)y=0$ near with and polynomials: substitute and its term-by-term derivatives; shift the index in each sum so that every sum runs over ; combine them into one series ; and set every to zero. For this gives .

\begin{aligned}&\sum_{n=2}^{\infty}n(n-1)a_{n}x^{n-2}\\&=\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^{n}\end{aligned}

The bookkeeping

Shifting an index changes nothing but the name: put $m=n-2$ , then rename $m$ as $n$ .
For $y''+y=0$ : , so .

A power series that is zero has zero coefficients

\sum_{n\ge0}c_{n}x^{n}=0\ \text{near}\ 0\ \Longrightarrow\ c_{n}=0\ \text{for every}\ n

If the series is zero on an interval about zero, so is every derivative of it. By the last lesson each coefficient is a derivative at zero divided by a factorial, so each coefficient is zero. That is why, after substituting, every coefficient of the combined series may be set to zero on its own.

Proof steps

f(x)=\sum c_{n}x^{n}=0\ \text{on an interval about}\ 0

Suppose the series vanishes near zero.

f^{(n)}(0)=0\ \text{for every}\ n

Every derivative of the zero function is zero.

c_{n}=\frac{f^{(n)}(0)}{n!}=0

The coefficients are the derivatives at zero divided by factorials.

Applications

Practice

Shift the Index

Rename the index so that every sum runs over the same power of x. Nothing changes but the label.

\begin{aligned}&\sum_{n=2}^{\infty}n(n-1)a_{n}x^{n-2}\\&=\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^{n}\end{aligned}

Try it

Written as a sum over $x^{n}$ from $n=0$ , what is $\sum_{n\ge1}na_{n}x^{n-1}$ ?