# Numerical Methods for First Order ODEs

Here we consider numerical methods to solve differential equations of the form

with a given intial condition .

In particular, we will look at the Forward Euler, Backward Euler, Trapezoid, Runge-Kutta 4, Adam-Bashforth 2 and Heun methods.

## Contents

## Forward Euler

The formula is:

where is the step-size.

## Backward Euler

The formula is:

where is the step-size.

## Trapezoidal

The formula is:

where is the step-size.

## Runge-Kutta 4

This is the following 4th-order Runge-Kutta method:

with

,

,

,

.

where is the step-size.

## Adam-Bashforth 2

This is a two-step method with the following formula:

where is the step-size.

## Heun

Heun's method has the following formula:

.

where is the step-size. So in Heun's method, there is a first approximation at the intermediate value, and then the final approximation at the next integration point.

## The GUI

Running the downloadable MATLAB® code on this page opens a GUI which allows you to compare different numerical methods for first order ODEs.

IB_ODE_GUI

@(x)(func(x))

## The Code

- IB_ODE_GUI.m (Run this)
- IB_ODE_GUI.fig (Required)

All files as zip archive: ode_all.zip