Euler’s Method

You’ll be able to

Approximate a solution numerically using Euler’s method
Carry out repeated steps with a fixed step size
Explain why Euler’s method produces an approximation, not an exact value

Following the tangent line, step by step

Euler’s method approximates a solution to dy/dx = f(x, y) when you cannot (or need not) solve it exactly. Starting from a known point, it uses the differential equation to get the slope there, then steps forward a small distance Δx along that tangent line to estimate the next point. It repeats: new slope, new step, new point. Each step is a straight-line approximation of a curve, so the method literally walks along the slope field.

Euler’s method update

x_{n+1} = x_n + Δx · y_{n+1} = y_n + f(x_n, y_n)·Δx

The new y is the old y plus slope times step size. "Rise = slope × run" applied one step at a time.

Why it is only an approximation

Because each step assumes the slope stays constant across the whole interval Δx — when it actually changes — Euler’s method accumulates error. Smaller steps track the true curve more closely but require more computation. For a solution that is concave up, the tangent lines lie below the curve, so Euler’s method underestimates; for a concave down solution it overestimates. Knowing the concavity lets you predict the direction of the error.

Worked example

Use Euler’s method with Δx = 0.5 to approximate y(1), given dy/dx = x + y and y(0) = 1.

1.Start at (x₀, y₀) = (0, 1). Slope there: f(0, 1) = 0 + 1 = 1.
2.Step one: y₁ = y₀ + slope·Δx = 1 + (1)(0.5) = 1.5, at x₁ = 0.5.
3.New slope at (0.5, 1.5): f = 0.5 + 1.5 = 2.
4.Step two: y₂ = 1.5 + (2)(0.5) = 2.5, at x₂ = 1. So y(1) ≈ 2.5.

Answer: y(1) ≈ 2.5 after two steps. Because the true solution is concave up here, this Euler estimate is an underestimate.

Watch out

Recompute the slope at the start of every step using the newest (x, y). A frequent error is reusing the initial slope for all steps — Euler’s method updates the slope each time before stepping forward.

Checkpoint

Euler’s method is used with dy/dx = 2x, starting at (1, 4) with step size Δx = 0.1. What is the approximate y-value at x = 1.1?

Checkpoint

If the true solution curve is concave up, how does Euler’s method’s approximation compare to the actual value?

On the exam

Organize Euler’s method in a small table with columns for x, y, slope f(x,y), and Δy = slope·Δx. The table structure prevents arithmetic slips across steps and is exactly what free-response graders expect to see.

Answer the 2 checkpoints as you read.