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

Euler's method approximates the solution of a first-order differential equation by repeatedly moving a short distance along the tangent at the current point. For $\frac{dy}{dx}=f(x,y)$, use $x_{n+1}=x_n+h$ and $y_{n+1}=y_n+h f(x_n,y_n)$. This gives a numerical approximation rather than an exact solution.

This is Euler's method, part of IB Mathematics: Analysis and Approaches HL, AHL 5.18 Differential equations. The differential equation supplies the current gradient, while a chosen step size $h$ controls how far the approximation moves.

At the current point $(x_n,y_n)$, the differential equation gives the slope:

$$m_n=f(x_n,y_n).$$

Over one horizontal step $h$, the estimated vertical change is $h f(x_n,y_n)$. Therefore,

$$y_{n+1}=y_n+h f(x_n,y_n).$$

Both coordinates must be updated from the same current point before the next gradient is calculated.

Consider $\frac{dy}{dx}=x+y$ with $y(0)=1$. Using , estimate . Two iterations are required because . Before starting, check that the target x-value can be reached by whole steps of size ; otherwise, the final step size must be adjusted.

Hence, Euler's method gives $y(0.2)\approx1.22$. The exact solution is not needed; the question tests correct use of the recurrence.

A smaller value of $h$ usually improves accuracy because shorter tangent segments follow a curved solution more closely. However, more iterations are required, and Euler's method still accumulates error. If asked to compare approximations, explain this trade-off rather than merely stating that “smaller is better”.

A common misconception is to evaluate every gradient at the initial point. Instead, use the latest point $(x_n,y_n)$ at every iteration. Another error is mixing $x_{n+1}$ with inside the slope; evaluate first, then update both values.

Exam technique: This AHL topic may appear in analytic form on Paper 1 or with technology on HL Papers 2 and 3. For calculate or find, state the recurrence, show the required iterations, and round only at the end to the requested accuracy.