Calculus II · Unit 4B · lesson

A Preview of Series Solutions to Differential Equations

Concept

Learning objectives

use coefficient matching to see how a differential equation can determine a power series recursively.

A Preview of Series Solutions to Differential Equations

Explanation

A differential equation can solve for coefficients one at a time

Suppose a solution is represented by $y=\sum c_nx^n$ . Differentiating the series and substituting into a differential equation turns the equation into an identity between power series. Equal power series have equal coefficients, so the differential equation generates recurrence relations among the $c_n$ .

This method extends beyond equations with elementary closed forms and leads to special functions. In this unit we use it only as a preview, because a full treatment belongs with differential equations. The important idea is that an infinite function problem becomes a systematic algebra problem in coefficients.

Bridge

An unknown function can be replaced by unknown coefficients

To solve a differential equation by power series, assume the solution has the form $y=\sum a_nx^n$ . Differentiate term by term, substitute the series into the equation, align powers, and compare coefficients. The differential equation becomes a recurrence for the numbers $a_n$ .

This method extends polynomial reasoning rather than introducing magic. Initial conditions determine starting coefficients, and the recurrence determines the rest. A full theory must also prove convergence, but the coefficient mechanism already previews why power series are central in differential equations and mathematical physics.

**A differential equation becomes a recurrence**
A flow diagram starts with the initial coefficient a0 and repeatedly applies the next-coefficient recurrence, producing factorial denominators in succession.
A differential equation becomes a recurrence. A coefficient machine mapping a n to a n+1 .

Proof idea

Equal power series have equal coefficients

Inside a common convergence interval, if two power series represent the same function, differentiating repeatedly at the center isolates each coefficient. This uniqueness lets the differential equation determine a recurrence term by term.

Concept

Coefficient matching

If two power series agree on an interval,

\sum a_nx^n=\sum b_nx^n,

then $a_n=b_n$ for every $n$ .

Guided walkthrough

Let $y=\sum_{n=0}^{\infty}c_nx^n$ . Then

y'=\sum_{n=0}^{\infty}(n+1)c_{n+1}x^n.

The equation $y'=y$ gives $(n+1)c_{n+1}=c_n$ , so $c_n=c_0/n!$ . Therefore

y=c_0\sum_{n=0}^{\infty}\frac{x^n}{n!}=c_0e^x.

Worked example

Recover the exponential from y prime equals y

Assume

y=\sum_{n=0}^{\infty}a_nx^n.

Then

y'=\sum_{n=0}^{\infty}(n+1)a_{n+1}x^n.

The equation $y'=y$ gives

(n+1)a_{n+1}=a_n, \qquad a_{n+1}=\frac{a_n}{n+1}.

If $y(0)=1$ , then $a_0=1$ , so $a_n=1/n!$ and

y=\sum_{n=0}^{\infty}\frac{x^n}{n!}=e^x.

Common mistake

Align the index before comparing coefficients

The series for $y'$ initially begins with $x^{n-1}$ . Reindex it so both sides use the same power $x^n$ before equating coefficients.

Interactive checku4b-series_solutions_preview-01

If $(n+1)c_{n+1}=c_n$ , express $c_n$ in terms of $c_0$ .

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Show hint

Compute the first few coefficients.

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Exercise

Use a series to analyze $y^{\prime\prime}+y=0$ .

Exercise

Explain how initial conditions determine free coefficients.

Exercise

Why must indices be shifted before coefficient matching?

Exercise

Name one reason series solutions are useful.

After the explanation

Use the section idea

Reading lens

Pair every Taylor approximation with a degree, center, target input, and certified remainder bound.

Mental model

The polynomial supplies the estimate; the remainder theorem supplies the trust boundary.

Decision

Choose a tractable center and degree, bound the needed derivative on the whole interval, then compare the bound with the required tolerance.

Common trap

Evaluating the next term without checking that the theorem's hypotheses make it a valid error bound.

Check yourself

Can you state exactly why the reported digits are certified?

Source & rights

Original instruction with traceable references.

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