AHL 5.16—Eulers method for 1st order DEs

Euler's Method: A numerical technique used to find approximate solutions to first-order ordinary differential equations (ODEs) given an initial condition. Rather than solving the ODE analytically, it uses the slope at each point to step forward to the next estimate.

Euler's method is named after the prolific Swiss mathematician Leonhard Euler. It is particularly valuable when an ODE has no neat closed-form solution , which happens more often in real-world modelling than you might think.

The core idea is beautifully simple: if you know where you are on the solution curve and you know the slope of the curve at that point, you can take a small step in the direction of that slope to estimate where the curve goes next. Repeat this process many times, and you trace out an approximate solution.

For a first-order ODE of the form:

$\frac{dy}{dx}=f(x,y)$

with an initial condition $y(x_0)=y_0$, Euler's method generates successive approximations using:

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

and the $x$-values are updated simply by:

$x_{n+1}=x_n+h$

where:

• $y_n$ is the current approximate $y$-value
• $x_n$ is the current $x$-value
• $h$ is the step size (chosen by the user)
• $f(x_n,y_n)$ is the slope of the solution curve at $(x_n,y_n)$