NUMERICAL METHODS GUIDE

How Euler's method estimates a differential equation

Euler's method advances an initial value problem one fixed step at a time by using the current slope. It is simple, transparent, and useful for checking how step size affects a numerical approximation.

FORMULA

yₙ₊₁ = yₙ + h · f(xₙ, yₙ)

WORKED EXAMPLE

For y′ = y with y(0) = 1 and h = 0.1, Euler's method uses the current value of y as the slope, then repeats the update across the selected x interval.

STEP BY STEP

Write the initial value problem

Start with a right-hand side such as y′ = f(x, y), an initial x value, and an initial y value.

Choose a fixed step size

The step h controls how far the method advances at each update. Smaller steps usually improve accuracy but require more calculations.

Evaluate the current slope

At the current point, calculate f(xₙ, yₙ). This is the slope used for the next local estimate.

Update the state

Add h multiplied by the slope to yₙ, move x forward by h, and repeat until the final x value is reached.

Check the approximation

Compare a smaller step size or a higher-order method such as Heun or RK4 before relying on the result.