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Euler's Method
Numerical Approximation

Learn to approximate solutions to differential equations step by step when exact solutions are out of reach.

AP Calculus BC

What Is Euler's Method?

Many differential equations of the form \(\frac{dy}{dx} = f(x, y)\) cannot be solved with an exact, closed-form formula. Euler's Method gives us a way to numerically approximate the solution by stepping along the tangent line at each point.

Visual Intuition: Walking Along Tangent Lines

x y True solution Euler approx. Start Step 1 Step 2 Step 3 Step 4

Why Does Euler's Method Matter?

Unsolvable DEs
Most real-world differential equations have no closed-form solution. Numerical methods are the only option.
Foundation
Euler's Method is the simplest numerical method and the gateway to more advanced techniques like Runge-Kutta.
AP Exam
Tested on both FRQ and MCQ. You must execute the algorithm by hand with a table of values.

The Core Formula

Given a differential equation \(\frac{dy}{dx} = f(x, y)\) with initial condition \(y(x_0) = y_0\) and step size \(\Delta x\):

In words: the new \(y\) equals the old \(y\) plus the slope at the old point times the step size.

One Step Up Close

\(\Delta x\) slope · \(\Delta x\) \((x_n, y_n)\) \((x_{n+1}, y_{n+1})\) True value Error \(x_n\) \(x_{n+1}\)

Euler's Method vs. Exact Solutions

Euler's Method Separation of Variables
Output Approximate \(y\) values at specific points Exact function \(y(x)\)
When to use DE is not separable or exact solution is impractical DE can be written as \(g(y)\,dy = h(x)\,dx\)
Accuracy Depends on \(\Delta x\): smaller is better Perfectly accurate
Requires Initial condition + \(f(x,y)\) + step size Integrability of both sides

How to Apply Euler's Method

1
Identify the pieces. Write down \(f(x,y)\), the initial condition \((x_0, y_0)\), the step size \(\Delta x\), and the target \(x\)-value.
2
Evaluate the slope. Compute \(f(x_n, y_n)\) at the current point.
3
Step forward. Calculate \(x_{n+1} = x_n + \Delta x\) and \(y_{n+1} = y_n + f(x_n, y_n) \cdot \Delta x\).
4
Repeat. Use the new point as your starting point and go back to Step 2 until you reach the target \(x\).
5
Report. State the approximate \(y\)-value at the target \(x\). Note whether it is an over- or underestimate if asked.

Worked Example 1: Two Steps

Given \(\frac{dy}{dx} = x + y\), \(y(0) = 1\), and \(\Delta x = 0.5\). Approximate \(y(1)\).

Step \(x_n\) \(y_n\) \(f(x_n, y_n)\) \(y_{n+1}\)
0 0 1 0 + 1 = 1 1 + 1(0.5) = 1.5
1 0.5 1.5 0.5 + 1.5 = 2 1.5 + 2(0.5) = 2.5

Example 1: Visualized

0 0.5 1.0 1 2 3 Exact curve (0, 1) (0.5, 1.5) (1, 2.5)

Worked Example 2: Smaller Steps

Same DE: \(\frac{dy}{dx} = x + y\), \(y(0)=1\), now with \(\Delta x = 0.25\). Approximate \(y(1)\).

n \(x_n\) \(y_n\) \(f(x_n, y_n)\) \(y_{n+1}\)
0 0 1 1 1.25
1 0.25 1.25 1.5 1.625
2 0.5 1.625 2.125 2.15625
3 0.75 2.15625 2.90625 2.8828

Worked Example 3: Non-Linear DE

Given \(\frac{dy}{dx} = y - x^2\), \(y(0) = 0.5\), \(\Delta x = 0.2\). Approximate \(y(0.4)\).

n \(x_n\) \(y_n\) \(f(x_n, y_n) = y_n - x_n^2\) \(y_{n+1}\)
0 0 0.5 \(0.5 - 0 = 0.5\) \(0.5 + 0.5(0.2) = 0.6\)
1 0.2 0.6 \(0.6 - 0.04 = 0.56\) \(0.6 + 0.56(0.2) = 0.712\)

Over- or Underestimate?

Concave Up (\(y'' > 0\))
Tangent lines lie below the curve. Euler's Method gives an underestimate.
Concave Down (\(y'' < 0\))
Tangent lines lie above the curve. Euler's Method gives an overestimate.

Common Mistakes

Using the new y for slope
The slope \(f(x_n, y_n)\) must be computed at the old point. Do not substitute \(y_{n+1}\) back into \(f\) during the same step.
Wrong step size
If asked to take 3 steps from \(x=1\) to \(x=2.5\), then \(\Delta x = \frac{2.5 - 1}{3} = 0.5\). Always calculate \(\Delta x\) from the given information.
Forgetting to update x
Both \(x\) and \(y\) advance each step. If \(f\) depends on \(x\), using the old \(x\) in subsequent steps produces wrong slopes.

Quick Reference Summary

Concept Key Fact
Formula \(y_{n+1} = y_n + f(x_n, y_n) \cdot \Delta x\)
Step Size Smaller \(\Delta x\) = better accuracy, more computation
Concave up Euler underestimates
Concave down Euler overestimates
Requires \(f(x,y)\), initial condition, \(\Delta x\)
Error per step Proportional to \((\Delta x)^2\) (local truncation error)

Question 1 of 5

Given \(\frac{dy}{dx} = 2x\), \(y(1) = 3\), and \(\Delta x = 0.5\), what is the Euler approximation for \(y(1.5)\)?

Question 2 of 5

In Euler's Method, the slope at each step is evaluated using which values?

Question 3 of 5

If the true solution to a DE is concave up on the interval of approximation, Euler's Method will produce:

Question 4 of 5

Given \(\frac{dy}{dx} = y\), \(y(0) = 1\), and \(\Delta x = 1\). Using one Euler step, what is \(y(1)\)?

Question 5 of 5

You need to approximate \(y(3)\) starting from \(y(1) = 2\) using Euler's Method with 4 equal steps. What is \(\Delta x\)?