Maclaurin Series from Differential Equations

students, imagine trying to describe a curved path using only straight-line ideas 📈. In calculus, one powerful way to do this is with a Maclaurin series, which turns a function into an infinite polynomial centered at $x=0$. In this lesson, you will see how differential equations can help us create Maclaurin series, not just use them.

What you will learn

By the end of this lesson, you should be able to:

explain what a Maclaurin series is and why it matters
use a differential equation to build a series solution step by step
connect derivatives at $x=0$ to the coefficients of a Maclaurin series
recognize how this idea fits into wider calculus topics like differential equations, approximation, and modelling
use examples to justify each step of the method

This topic is important because many real-world systems are described by differential equations, but exact solutions are not always easy to find. A series solution gives a practical way forward, especially when the function itself is not simple enough to solve directly.

Maclaurin series: the basic idea

A Maclaurin series is a special Taylor series centered at $x=0$. It has the form

$f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f'''(0)}{3!}x^3+\cdots$

More compactly,

$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n.$

Each coefficient depends on a derivative of the function at $x=0$. So if you know enough derivatives, you can build the series.

Why is this useful? Because complicated functions can sometimes be approximated by a few terms. For example, for small $x$,

$e^x\approx 1+x+\frac{x^2}{2}.$

That is already a very good approximation near $x=0$.

The important idea in this lesson is that a differential equation can tell us the derivatives of a function, and therefore its Maclaurin series. Instead of finding the function first, we can work directly from the equation.

Using a differential equation to find a series

Suppose a function $y(x)$ satisfies a differential equation. If we want a Maclaurin series, we assume

$y(x)=a_0+a_1x+a_2x^2+a_3x^3+\cdots=\sum_{n=0}^{\infty}a_nx^n.$

Then we differentiate term by term:

$y'(x)=a_1+2a_2x+3a_3x^2+4a_4x^3+\cdots$

and again if needed.

We substitute these series into the differential equation, then compare coefficients of the same powers of $x$. This creates a recurrence relation, which links one coefficient to the next.

Example 1: a simple first-order differential equation

Let

$y'=y,$

with initial condition

$y(0)=1.$

Assume

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

Then

$y'=\sum_{n=1}^{\infty}na_nx^{n-1}.$

It is often easier to rewrite this with matching powers of $x$:

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

Since $y'=y$,

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

So for each $n$,

$(n+1)a_{n+1}=a_n.$

This gives the recurrence

$a_{n+1}=\frac{a_n}{n+1}.$

The initial condition $y(0)=1$ means

$a_0=1.$

Now compute the first few terms:

$a_1=1,$

$\quad a_2=\frac{1}{2!},$

$\quad a_3=\frac{1}{3!},$

$\quad a_4=\frac{1}{4!}.$

So the series is

$y=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots$

which matches the Maclaurin series of $e^x$. That is a beautiful result ✨.

Building series from higher-order differential equations

Many IB-style problems involve second-order or higher-order equations. The process is the same: assume a series, differentiate as needed, substitute, and match coefficients.

Example 2: a second-order equation

Consider

$y''+y=0,$

with initial conditions

$y(0)=0, \quad y'(0)=1.$

Assume

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

Then

$y''=\sum_{n=2}^{\infty}n(n-1)a_nx^{n-2}.$

Rewrite $y''$ so it uses powers of $x^n$:

$y''=\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^n.$

Substitute into the differential equation:

$\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^n+\sum_{n=0}^{\infty}a_nx^n=0.$

Therefore,

$(n+2)(n+1)a_{n+2}+a_n=0,$

$a_{n+2}=-\frac{a_n}{(n+2)(n+1)}.$

Now use the initial conditions:

$a_0=y(0)=0,$

and

$y'(x)=a_1+2a_2x+3a_3x^2+\cdots,$

$y'(0)=a_1=1.$

Then

$a_2=-\frac{a_0}{2!}=0,$

$a_3=-\frac{a_1}{3\cdot 2}=-\frac{1}{3!},$

$a_4=0,$

$a_5=-\frac{a_3}{5\cdot 4}=\frac{1}{5!}.$

$y=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$

which matches the Maclaurin series of $\sin x$.

This example shows a major IB idea: differential equations can reveal a familiar function through its series expansion.

Why the recurrence relation matters

A recurrence relation is a rule that lets one coefficient generate the next. In series methods, this is extremely useful because it turns a differential equation into an algebra pattern.

For students, think of it like a chain of dominoes 🎯. Once you know the first piece, each new coefficient follows from the previous one.

The recurrence can help you:

compute several terms quickly
detect a pattern
compare the result with a known series
approximate the solution near $x=0$

Sometimes the recurrence produces separate patterns for even and odd powers. This often happens in second-order equations and is one reason why functions like $\sin x$ and $\cos x$ split naturally into odd and even terms.

Connecting series solutions to approximation and modelling

In calculus, exact answers are not always available. A series solution gives a local approximation near $x=0$.

For example, if a function satisfies a differential equation and the series begins

$y=1+2x-\frac{x^2}{3}+\cdots,$

then for small $x$ you can estimate $y$ by keeping only the first few terms.

This matters in physics and engineering. Differential equations describe motion, growth, vibration, and electrical circuits. If a solution cannot be written in a simple closed form, a Maclaurin series can still give useful numerical information.

For instance, near $t=0$, a solution to a motion model can be approximated to estimate position or velocity. In that setting, the derivative information from the differential equation gives a local description of the system, which is exactly what series methods are designed to do.

Common exam-style approach

When IB asks for a Maclaurin series from a differential equation, a good method is:

Assume $y=\sum_{n=0}^{\infty}a_nx^n$.
Differentiate to get $y'$, $y''$, or higher derivatives.
Substitute into the differential equation.
Match coefficients of the same powers of $x$.
Use initial conditions to find $a_0$, $a_1$, and more.
Write the first few terms of the series.
If possible, identify the function or comment on the approximation.

Be careful with shifting indices. This is a common source of mistakes. If one sum has $x^{n-1}$ and another has $x^n$, rewrite the first sum so the powers match before comparing coefficients.

Also remember that a Maclaurin series is centered at $x=0$, so the value and derivatives at $x=0$ are especially important.

Conclusion

Maclaurin series from differential equations combines two key calculus ideas: infinite series and differential equations. students, the main strategy is to assume a power series form, substitute it into the differential equation, and use coefficient matching to build a recurrence relation. With the initial conditions, you can determine the coefficients and write the Maclaurin series.

This method is valuable because it links abstract algebraic patterns with real functions and practical modelling. It also helps you see how calculus topics connect: differentiation gives the coefficients, integration and approximation help interpret the result, and differential equations provide the structure. In IB Mathematics: Analysis and Approaches HL, this is a strong example of how a general method can reveal deep properties of familiar and unfamiliar functions alike 🚀.

Study Notes

A Maclaurin series is a Taylor series centered at $x=0$.
The general form is $f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n$.
For series solutions, assume $y=\sum_{n=0}^{\infty}a_nx^n$.
Differentiate term by term to find $y'$, $y''$, and so on.
Substitute the series into the differential equation.
Match coefficients of like powers of $x$ to get a recurrence relation.
Use initial conditions such as $y(0)$ and $y'(0)$ to find starting coefficients.
A recurrence relation lets you generate more terms of the series.
The resulting series can approximate the solution near $x=0$.
This method connects differential equations, approximation, and modelling in calculus.