Level 7.4 – Differential equations

These notes include all proofs for stuff not typically proven in this course, as I refuse to use and not prove stuff!

These notes are based on lectures by Christopher Thomas in the academic 2025-2026. Mistakes are almost surely mine and this is not official nor a recreation of what was in the lectures.

Summary of the course [Expand]

• Lectures 1-3 are A level review. My notes include a proof of the mean value theorem, Rolle's theorem and Taylor's theorem.
• Lectures 4-5 are an introduction to multivariable calculus: Partial derivatives, directional derivatives, the multivariate chain rule, symmetry of mixed partials, and differentiating under the integral sign. My notes provide proofs (under appropriate conditions on the functions in question).
• Lectures 6-8 are mostly an A level review of differential equations stuff but we introduce some new perspectives.
• Lecture 8 introduces exact equations and in my notes I provide a proof of a sufficient condition for an equation to be exact.
• Lectures 8-10 are about the stability of solutions and how to analyze it.
• Lectures 11-15 are about higher order equations and wronskians and I provide a proof in my notes that for linear equations of order up to 2, under appropriate conditions, we have a number of linearly independent solutions equal to the order.
• Lectures 16-17 are about the dirac delta and heaviside step functions, and we take a detour to talk about a few recurrence relations.
• Lectures 18-19 are about series solutions, a certain theorem (which my notes provide a proof of) and the frobenius method.
• Lectures 20-21 are about multivariate functions and their countours and classifying their stationary points using the hessian matrix. My notes provide a proof of Sylvestor's criterion and the implicit function theorem, which are used in this section.
• Lectures 22-23 are about solving systems of differential equations using matrix methods or analyzing the stability of points.
• Lectures 23-24 are a brief introduction to partial differential equations and we show here how to solve the wave equation using the method of characteristics.

See my differential equations notes here