7.5 Approximating Solutions Using Euler's Method

What Euler's Method Does

Euler's Method approximates points on a solution curve to a differential equation using tangent line steps. Starting at a known point, you use the slope from dy/dx at that point to estimate the next point, moving in small increments of Δx. Repeating this process builds a piecewise-linear approximation of the actual solution curve.

The Iterative Formula

Given a starting point (x₀, y₀) and step size Δx, each new point is found using: x_new = x_old + Δx and y_new = y_old + (dy/dx)·Δx, where dy/dx is evaluated at (x_old, y_old). Repeat this process for each step to move further along the curve.

Accuracy and Limitations

Euler's Method gives only an approximation — the smaller the step size, the more accurate the estimate, but more steps are needed. If the actual solution is concave up, Euler's Method (using tangent lines) will underestimate the true value; if concave down, it will overestimate.

Key terms

Euler's Method

A numerical technique that approximates solutions to differential equations using tangent line steps from a known point.

Euler's Method formula

x_new = x_old + Δx; y_new = y_old + (dy/dx)·Δx, evaluated at (x_old, y_old).

Effect of smaller step size

Smaller Δx values increase accuracy but require more steps/calculations.

Concave up curve & Euler's Method

Produces an underestimate, since tangent lines lie below the curve.

Concave down curve & Euler's Method

Produces an overestimate, since tangent lines lie above the curve.

Try an AP-style question

dy/dx = x + y, y(0) = 1. Using Euler's method with step 0.1, what is the estimate of y(0.1)?

1. 1.2
2. 1.0
3. 1.01
4. 1.1