Euler's Method — Step-by-Step Approximation

Approximate solutions to differential equations using small steps.

Formula: y_new = y_old + (dy/dx) · Δx

Method:

Start with initial condition (x₀, y₀).

Calculate dy/dx at that point.

Step forward: x₁ = x₀ + Δx, y₁ = y₀ + (dy/dx)(Δx).

Repeat.

Example: dy/dx = x + y, y(0) = 1, Δx = 0.5, find y(1).

At (0,1): dy/dx = 0+1 = 1. y₁ = 1 + 1(0.5) = 1.5

At (0.5, 1.5): dy/dx = 0.5+1.5 = 2. y₂ = 1.5 + 2(0.5) = 2.5

So y(1) ≈ 2.5.

Smaller Δx = more accurate. Euler's method is a rough approximation — the AP exam tests whether you understand the process, not whether you get an exact answer.

Reference:

Wikipedia: Euler's Method

https://en.wikipedia.org/wiki/Euler_method