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
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.
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.