How Do You Use Euler's Method to Numerically Solve a First Order Differential Equation?

Use Euler's method by starting from a known point and repeatedly moving forward using the gradient supplied by the differential equation. For step size $h$, apply $x_{n+1}=x_n+h$ and $y_{n+1}=y_n+h f(x_n,y_n)$; each new point becomes the starting point for the next step.

This is AHL 5.16, so it is IB Mathematics: Applications and Interpretation HL-only content. The example has a short-response Paper 1 style, but Euler's method may appear on any HL paper; a GDC is available throughout.

Suppose $\frac{dy}{dx}=x+y$, $y(0)=1$, and estimate using . Here, , with . The is

$$y_{n+1}=y_n+0.1(x_n+y_n).$$

The corresponding updates are $x_1=0.1$ and $x_2=0.2$. Therefore, $y(0.2)$.

Geometrically, the method replaces the solution curve on each interval with a short tangent-line segment. At the current point, the differential equation gives the tangent gradient, so $h f(x_n,y_n)$ estimates the vertical change. This explains why every gradient must be evaluated before either coordinate is updated.

Euler's method gives a numerical approximation, not an exact solution. A smaller $h$ generally improves accuracy because the tangent better matches the curve over a shorter interval, although more iterations are required. For repeated calculations, enter the two recurrences into a spreadsheet or GDC table and retain full precision.

A common misconception is to calculate the next gradient using partially updated coordinates. Instead, evaluate $f(x_n,y_n)$ using both current coordinates first. Another error is performing the wrong number of steps: calculate $\frac{x_{\text{target}}-x_0}{h}$.

Before calculating, determine the number of iterations and organize the values consistently. A table with columns for $n$, $x_n$, $y_n$, $f(x_n,$, and reduces substitution errors. Check that the final -value equals the requested target before reporting the corresponding approximate -value in context.

Exam technique: Write both recurrence equations and show the first iteration explicitly, since this provides evidence for the method mark. Round only the final answer and use $\approx$ because Euler's method is approximate.