Introduction to Ordinary Differential Equations

“Since Newton, mankind has come to realise that the laws of physics are always expressed in the language of differential equations” – Steven Strogatz

In this section of the course we will learn how to solve ordinary differential equations. These are of vital importance to a physicist as they are at the base of classical and celestial mechanics via Newton’s equations. They are also used to model electrical circuits and radioactive decay, amongst other systems.

Broadly speaking, there are three ways to solve a differential equation: analytically, numerically (with a pre-existing function) or numerically (with a home-made function). Numerical approaches offer more flexibility but with the caveat that they are approximate and must be converged.

In this lesson we will use numerical approaches to solve ODE’s, with a strong focus on home-made numerical functions (rather than pre-made functions imported from a library). As such we will achieve a greater understanding of the underlying mathematics and approximations used - and it’s a great chance to practice our Python skills.

Before you begin

• Check that Python and Jupyter Notebook are installed
• Launch a Jupyter notebook

Please see the Setup page for more details.

External resources

📖 Textbook: Mark Newman’s Computational Physics sections 8.1, 8.2, 8.3, 8.4