Mathfolis

Euler's Method

Unit 7 · Differential Equations

Euler's Method (BC Only)

Euler's Method approximates a solution to dy/dx = f(x,y) numerically by taking successive linear steps of size h. Each step uses the slope at the current point to estimate the next y value. If the solution is concave up, Euler underestimates; if concave down, it overestimates.

Recurrence Formula

xn+1=xn+hx_{n+1} = x_n + h
yn+1=yn+f(xn, yn)⋅hy_{n+1} = y_n + f(x_n,\, y_n)\cdot h

Over/Underestimate

Solution concave up (y″ > 0) → Euler underestimates
Solution concave down (y″ < 0) → Euler overestimates
Caution: Always evaluate f at (xₙ, yₙ) — the current point — not the next point. This is the most common Euler's Method error.
Type 1

Two-Step Euler Approximation

Build a table step by step: compute f(xₙ, yₙ), then update xₙ₊₁ and yₙ₊₁.

Example 1

Approximate y(0.2) using Euler's Method with h = 0.1 for dy/dx = x − y, y(0) = 1.

Example 2

Approximate y(0.2) for dy/dx = y², y(0) = 1, with h = 0.1.

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Type 2

Over/Underestimate via Concavity

Determine whether Euler's Method gives an overestimate or underestimate by analyzing the second derivative of the solution.

Example 3

For dy/dx = y with y(0) = 1, use h = 1 to approximate y(2). Is this an over- or underestimate?

Example 4

If the solution to a DE is concave down on [0, 2], will Euler's Method overestimate or underestimate? Explain.

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