Concept preview

What you will be able to do

1. 01State Euler's method as a repeated tangent-line step $y_{n+1}=y_n+f(x_n,y_n)\,h$ and explain why it approximates the solution of an initial value problem.
2. 02Carry out several Euler steps from a given initial point with a fixed step size $h$, recomputing the slope at each new point and organising the work in a clean table.
3. 03Explain how the step size $h$ controls accuracy, and predict whether the approximation lands above or below the true solution for a concave-up curve.

An initial value problem gives you two things: a differential equation $\dfrac{dy}{dx}=f(x,y)$ that reports the slope of the solution at any point $(x,y)$, and a starting point $(x_0,\,y_0)$ the solution passes through. Euler's method uses these to march forward in small horizontal steps of a chosen width $h$, the step size. At each stage you are standing at a known point on your approximation, you read off the slope the differential equation prescribes there, and you follow that straight tangent line across one step of width $h$ to reach the next point.