Approximating Solutions Using Euler's Method (College Board AP® Calculus BC) : Study Guide

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 26 February 2025

Euler's method

What is Euler’s method?

Euler’s method is a numerical method for finding approximate solutions to first order differential equations
It treats the derivatives in the equation as being constant over short ‘steps’
- At each step you use a linear approximation to approximate the value at the next point
  - See the 'Approximating Values of a Function' study guide in the Linearization topic
The differential equation must be in the form $\frac{d y}{d x} = f^{'} (x, y)$
- i.e. $\frac{d y}{d x}$ is expressed in terms of $x$ and $y$
Start with a known value $y_{0}$ corresponding to a value $x_{0}$
- Then $y_{1} = y_{0} + ∆ x \cdot f^{'} (x_{0}, y_{0})$
  - $x_{1} = x_{0} + ∆ x$
  - $∆ x$ is the step size
- $y_{2} = y_{1} + ∆ x \cdot f^{'} (x_{1}, y_{1})$
  - $x_{2} = x_{1} + ∆ x$
- $y_{3} = y_{2} + ∆ x \cdot f^{'} (x_{2}, y_{2})$
  - $x_{3} = x_{2} + ∆ x$
- etc.

Graph showing Euler's method with points (x0, y0) to (x4, y4). Heavier black line for approximation, lighter black line for exact solution, and slopes of approximation line segments labelled. — **Example of an Euler's method approximation**

An Euler’s method approximation tends to wander away from the exact solution
- The accuracy can be improved by making the step size smaller

Examiner Tips and Tricks

If an exam question asks you how to improve an Euler’s method approximation, the answer will almost always involve decreasing the step size $∆ x$ .

How do I use Euler’s method with a first order differential equation?

STEP 1: Make sure your differential equation is in $\frac{d y}{d x} = f^{'} (x, y)$ form
- It is possible that only one of the variables will appear on the right-hand side of the equation
  - For example, the derivative may be in terms of $x$ only, $\frac{d y}{d x} = f' (x)$
  - The procedure to follow remains the same
STEP 2: Use the recursion equations $y_{n + 1} = y_{n} + ∆ x \cdot f^{'} (x_{n}, y_{n})$ and $x_{n + 1} = x_{n} + ∆ x$
- An exam question may tell you the correct value of $∆ x$ to use
  - or you may need to calculate it from other info provided
  - See the Worked Examples
STEP 3: Start with $n = 0$
- This gives $y_{1} = y_{0} + ∆ x \cdot f^{'} (x_{0}, y_{0})$ and $x_{1} = x_{0} + ∆ x$
- The values for $x_{0}$ and $y_{0}$ will come from the initial conditions given in the question
- Your values for $x_{1}$ and $y_{1}$ will be carried over into the next step
STEP 4: Continue with $n = 1$
- This gives $y_{2} = y_{1} + ∆ x \cdot f^{'} (x_{1}, y_{1})$ and $x_{2} = x_{1} + ∆ x$
- The values for $x_{1}$ and $y_{1}$ will come from the previous step
- The value you find for $y_{2}$ will usually be the Euler's method approximation you are looking for

Examiner Tips and Tricks

It is possible to perform Euler's method with any step size and any number of steps. In this case you could go on to calculate $y_{3}$ , $y_{4}$ , $y_{5}$ , etc. On the exam, however, the questions almost always involve only two steps.

Examiner Tips and Tricks

You might find it useful to do your workings in a table

$n$	$x_{n}$	$y_{n}$
$0$	$x_{0}$	$y_{0}$
$1$	$x_{1} = x_{0} + Δ x$	$y_{1} = y_{0} + ∆ x \cdot f^{'} (x_{0}, y_{0})$
$2$	$x_{2} = x_{1} + Δ x$	$y_{2} = y_{1} + ∆ x \cdot f^{'} (x_{1}, y_{1})$

Worked Example

Let $y = f (x)$ be the solution to the differential equation $\frac{d y}{d x} = x - 3 y + 1$ with initial condition $f (0) = 2$ . What is the approximation for $f (1)$ obtained by using Euler's method with a step size of $\frac{1}{2}$ starting at $x = 0$ ?

You need to go from $x = 0$ to $x = 1$ with a step size of $\frac{1}{2}$

so there will be $\frac{1 - 0}{\frac{1}{2}} = 2$ steps

Use the formula $y_{n + 1} = y_{n} + ∆ x \cdot f^{'} (x_{n}, y_{n})$

where here $∆ x = \frac{1}{2}$
and $\frac{d y}{d x} = x - 3 y + 1$

Start with initial value $(x_{0}, y_{0}) = (0, 2)$

$(x_{0}, y_{0}) = (0, 2)$

Then $x_{1} = x_{0} + Δ x$

$x_{1} = 0 + \frac{1}{2} = \frac{1}{2}$

And $y_{1} = y_{0} + ∆ x \cdot f^{'} (x_{0}, y_{0})$

$y_{1} = 2 + \frac{1}{2} \cdot ((0) - 3 (2) + 1) = 2 + \frac{1}{2} (- 5) = - \frac{1}{2}$

$\Rightarrow (x_{1}, y_{1}) = (\frac{1}{2}, - \frac{1}{2})$

Continue from $(x_{1}, y_{1}) = (\frac{1}{2}, - \frac{1}{2})$

$x_{2} = x_{1} + Δ x$

$x_{2} = \frac{1}{2} + \frac{1}{2} = 1$

And $y_{2} = y_{1} + ∆ x \cdot f^{'} (x_{1}, y_{1})$

$y_{2} = - \frac{1}{2} + \frac{1}{2} \cdot ((\frac{1}{2}) - 3 (- \frac{1}{2}) + 1) = - \frac{1}{2} + \frac{1}{2} (3) = 1$

$\Rightarrow (x_{2}, y_{2}) = (1, 1)$

You could also do these workings in a table

$n$	$x_{n}$	$y_{n}$
$0$	$0$	$2$
$1$	$0 + \frac{1}{2} = \frac{1}{2}$	$\begin{array}{rcl} 2 + \frac{1}{2} \cdot ((0) - 3 (2) + 1) & = & 2 + \frac{1}{2} (- 5) \\ = & - \frac{1}{2} \end{array}$
$2$	$\frac{1}{2} + \frac{1}{2} = 1$	$\begin{array}{rcl} - \frac{1}{2} + \frac{1}{2} \cdot ((\frac{1}{2}) - 3 (- \frac{1}{2}) + 1) & = & - \frac{1}{2} + \frac{1}{2} (3) \\ = & 1 \end{array}$

So by the Euler method, $f (1) \approx y_{2} = 1$

$f (1) \approx 1$

Worked Example

$x$	2	2.5	3
$f^{'} (x)$	5	2	1

$f$ is a twice-differentiable function for all values of $x$ , with $f (2) = 1$ . The table above gives values of the derivative of $f$ , $f^{'}$ , for selected values of $x$ .

Use Euler's method, with two steps of equal size starting at $x = 2$ , to approximate $f (3)$ . Show the computations that lead to your answer.

You need to go from $x = 2$ to $x = 3$ with two steps of equal size

so the step size will be $Δ x = \frac{3 - 2}{2} = \frac{1}{2}$

Use the formula $y_{n + 1} = y_{n} + ∆ x \cdot f^{'} (x_{n}, y_{n})$

where here $\frac{d y}{d x} = f' (x)$
i.e. where the value of $\frac{d y}{d x}$ depends on $x$ only

Start with initial value $(x_{0}, y_{0}) = (2, 1)$

$(x_{0}, y_{0}) = (2, 1)$

Then $x_{1} = x_{0} + Δ x$

$x_{1} = 2 + \frac{1}{2} = \frac{5}{2}$

And $y_{1} = y_{0} + ∆ x \cdot f^{'} (x_{0})$

$y_{1} = 1 + \frac{1}{2} \cdot f^{'} (2) = 1 + \frac{1}{2} (5) = \frac{7}{2}$

$\Rightarrow (x_{1}, y_{1}) = (\frac{5}{2}, \frac{7}{2})$

Continue from $(x_{1}, y_{1}) = (\frac{5}{2}, \frac{7}{2})$

$x_{2} = x_{1} + Δ x$

$x_{2} = \frac{5}{2} + \frac{1}{2} = 3$

And $y_{2} = y_{1} + ∆ x \cdot f^{'} (x_{1})$

$y_{2} = \frac{7}{2} + \frac{1}{2} \cdot f^{'} (2.5) = \frac{7}{2} + \frac{1}{2} (2) = \frac{9}{2}$

$\Rightarrow (x_{2}, y_{2}) = (3, \frac{9}{2})$

You could also do these workings in a table

$n$	$x_{n}$	$y_{n}$
$0$	$2$	$1$
$1$	$2 + \frac{1}{2} = \frac{5}{2}$	$\begin{array}{rcl} 1 + \frac{1}{2} \cdot f^{'} (2) & = & 1 + \frac{1}{2} (5) \\ = & \frac{7}{2} \end{array}$
$2$	$\frac{5}{2} + \frac{1}{2} = 3$	$\begin{array}{rcl} \frac{7}{2} + \frac{1}{2} \cdot f^{'} (2.5) & = & \frac{7}{2} + \frac{1}{2} (2) \\ = & \frac{9}{2} \end{array}$

So by the Euler method, $f (3) \approx y_{2} = \frac{9}{2}$

$f (3) \approx \frac{9}{2}$

How can I tell if an Euler's method approximation is an overestimate or an underestimate?

If the solution to a differential equation is concave down on the interval over which Euler's method is employed
- then the Euler's method approximation will be an overestimate
- See the diagram at the start of this study guide for an example of this
If the solution to a differential equation is concave up on the interval over which Euler's method is employed
- then the Euler's method approximation will be an underestimate
You can differentiate $\frac{d y}{d x}$ again with respect to $x$ to find $\frac{d^{2} y}{d x^{2}}$
- Then $\frac{d^{2} y}{d x^{2}} < 0$ tells you the solution is concave down
- And $\frac{d^{2} y}{d x^{2}} > 0$ tells you the solution is concave up
- Note that you don't need to solve the equation to determine its concavity!

How can I use my graphing calculator to carry out Euler’s method?

STEP 1: Make sure your differential equation is in $\frac{d y}{d x} = f^{'} (x, y)$ form
STEP 2: Write down the recursion equations $y_{n + 1} = y_{n} + ∆ x \cdot f^{'} (x_{n}, y_{n})$ and $x_{n + 1} = x_{n} + ∆ x$
- An exam question may tell you the correct value of $∆ x$ to use
  - or you may need to calculate it from other info provided
  - See the Worked Example
STEP 3: Use the recursion feature on your calculator to calculate the Euler’s method approximation over the correct number of steps
- The values for $x_{0}$ and $y_{0}$ will come from the initial conditions given in the question
- Your calculator will output the answer as a table of values

Examiner Tips and Tricks

Be careful with letters. In the exam (and in your calculator’s recursion app) the variables may not be x and y.

Examiner Tips and Tricks

Although your graphing calculator can carry out Euler's method approximations, these questions are usually on the non-calculator part of the exam. So make sure you can do the method 'by hand' as well!

Worked Example

Consider the differential equation $\frac{d y}{d x} + y = x + 1$ with the initial condition $y (0) = 0.5$ .

(a) Using your graphing calculator, apply Euler’s method with five equal steps to approximate the solution to the differential equation at $x = 1$ .

First rearrange into $\frac{d y}{d x} = f^{'} (x, y)$ form

$\frac{d y}{d x} = x - y + 1$

Next calculate the step size $∆ x$ ; you need to go from $x = 0$ to $x = 1$ in five steps so

$∆ x = \frac{1 - 0}{5} = 0.2$

Put that information into the recursion equations $y_{n + 1} = y_{n} + ∆ x \cdot f^{'} (x_{n}, y_{n})$ and $x_{n + 1} = x_{n} + ∆ x$

$y_{n + 1} = y_{n} + 0.2 \cdot (x_{n} - y_{n} + 1)$

$x_{n + 1} = x_{n} + 0.2$

Put those equations into the recursion app on your calculator, and specify the required number of steps ( $n = 5$ )

Your calculator will output the answers in a table

$n$	$x_{n}$	$y_{n}$
0	0	0.5
1	0.2	0.6
2	0.4	0.72
3	0.6	0.856
4	0.8	1.0048
5	1	1.1638

Note that the last row in the table gives the result you're looking for (i.e. the Euler approximation for $y$ when $x = 1$ )

Round to 3 decimal places

$y (1) \approx 1.164 (3 d . p .)$

(b) Explain how the accuracy of the approximation in part (a) could be improved.

Decrease the step size by using a greater number of steps to get from 0 to 1

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Approximating Solutions Using Euler's Method (College Board AP® Calculus BC) : Study Guide

Euler's method

What is Euler’s method?

How do I use Euler’s method with a first order differential equation?

How can I tell if an Euler's method approximation is an overestimate or an underestimate?

How can I use my graphing calculator to carry out Euler’s method?

You've read 0 of your 5 free study guides this week

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Join the 100,000+ Students that ❤️ Save My Exams

Unit 1: Limits & Continuity

Limits

Definition of a Limit

Infinite Limits & Limits at Infinity

Properties of Limits

Evaluating Limits Analytically

Evaluating Limits Numerically & Graphically

Squeeze Theorem & Trigonometric Limits

Summary of Limits

Selecting Procedures for Determining Limits

Continuity

Continuity

Removable Discontinuities

Non-removable Discontinuities

Intermediate Value Theorem

Unit 2: Differentiation Definition & Fundamental Properties

Definition of Differentiation

Average Rate of Change

Instantaneous Rate of Change

Derivatives & Tangents

Estimating the Derivative at a Point

Differentiability & Continuity

Fundamental Properties of Differentiation

Derivative Rules

Derivatives of Exponentials and Logarithms

Derivatives of Sine and Cosine Functions

The Product Rule

The Quotient Rule

Derivatives of Tangent and Reciprocal Trig Functions

Unit 3: Differentiation of Composite, Implicit & Inverse Functions

Differentiation of Composite & Inverse Functions

The Chain Rule

The Inverse Function Theorem

Derivatives of Inverse Trigonometric Functions

Higher-Order Derivatives

Summary of Differentiation

Selecting Procedures for Calculating Derivatives

Unit 4: Contextual Applications of Differentiation

Rates of Change & Related Rates

Meaning of a Derivative in Context

Motion in a Straight Line

Related Rates

Linearization

Approximating Values of a Function

L'Hospital's Rule

L'Hospital's Rule

Unit 5: Analytical Applications of Differentiation

Graphs of Functions & Their Derivatives

Mean Value Theorem

Extreme Value Theorem

Critical Points

Increasing & Decreasing Functions

First Derivative Test for Local Extrema

Candidates Test for Global Extrema

Concavity of Functions

Second Derivative Test for Local Extrema

Graphs of f, f' & f''

Optimization Problems

Behaviors of Implicit Relations

Second Derivatives of Implicit Functions

Critical Points of Implicit Relations

Unit 6: Integration & Accumulation of Change

Integration & Antiderivatives

Indefinite Integrals

Derivatives & Antiderivatives

Indefinite Integral Rules

Constant of Integration

Riemann Sums & Definite Integrals

Accumulation of Change

Riemann Sums

Trapezoidal sums