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Euler's Method — Definition, Formula & Examples

Euler's Method is a numerical technique that approximates the solution to a differential equation by taking small steps along the tangent line at each point. You start from a known initial value and repeatedly use the slope to estimate the next y-value.

Given an initial value problem dydx=f(x,y)\frac{dy}{dx} = f(x, y) with y(x0)=y0y(x_0) = y_0, Euler's Method generates a sequence of approximations yn+1=yn+h⋅f(xn,yn)y_{n+1} = y_n + h \cdot f(x_n, y_n), where hh is a fixed step size and xn+1=xn+hx_{n+1} = x_n + h. The method is a first-order numerical scheme whose accuracy improves as h→0h \to 0.

Key Formula

yn+1=yn+h⋅f(xn, yn)y_{n+1} = y_n + h \cdot f(x_n,\, y_n)
Where:
  • yn+1y_{n+1} = Approximate y-value at the next step
  • yny_n = Current approximate y-value
  • hh = Step size (the increment in x)
  • f(xn,yn)f(x_n, y_n) = Slope given by the differential equation at the current point

How It Works

You begin with a point (x0,y0)(x_0, y_0) where the solution is known. At that point, the differential equation gives you the slope f(x0,y0)f(x_0, y_0). Multiply the slope by the step size hh to find the change in yy, then add it to the current yy-value. Move to the new point (x1,y1)(x_1, y_1) and repeat. Each step follows the tangent line for a short distance, so smaller step sizes generally produce more accurate results.

Worked Example

Problem: Use Euler's Method with step size h = 0.5 to approximate y(1) given dy/dx = 2x and y(0) = 1.
Step 1: Start at (x₀, y₀) = (0, 1). Compute the slope: f(0, 1) = 2(0) = 0.
y1=1+0.5⋅0=1y_1 = 1 + 0.5 \cdot 0 = 1
Step 2: Move to x₁ = 0.5 with y₁ = 1. Compute the slope: f(0.5, 1) = 2(0.5) = 1.
y2=1+0.5⋅1=1.5y_2 = 1 + 0.5 \cdot 1 = 1.5
Answer: The Euler's Method approximation gives y(1) ≈ 1.5. (The exact solution is y = x² + 1, so y(1) = 2. The error shrinks with smaller step sizes.)

Why It Matters

Euler's Method appears on the AP Calculus BC exam (Unit 7) and is tested in free-response questions where you must approximate a solution from a given slope field or differential equation. Engineers and scientists use refined versions of this technique to simulate systems—from projectile trajectories to population models—when exact solutions are impossible to find.

Common Mistakes

Mistake: Using f(x_{n+1}, y_n) instead of f(x_n, y_n) for the slope
Correction: Always evaluate the slope at the current point (xₙ, yₙ), not at the next x-value. The whole idea is to use the slope where you are now to step forward.

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