Cambridge Part IA Differential Equations: Syllabus and Practice

Differential Equations is one of the four Michaelmas Term courses in Part IA, the first year of the Cambridge Mathematical Tripos. It has 24 lectures and is examined in Paper 2, alongside Probability. It begins with a fast review of calculus and then spends most of its time on ordinary differential equations. This guide follows the official 2026-27 Faculty of Mathematics schedules and links every topic to free practice.

What Cambridge Differential Equations covers

The schedules split the course into six blocks, with approximate lecture counts:

1. Basic calculus [3]. An informal treatment of differentiation as a limit, the chain rule, Leibniz's rule, Taylor series, O and o notation and l'Hôpital's rule, then integration as area, the Fundamental Theorem of Calculus, and substitution and parts.
2. Partial derivatives [2]. Geometric meaning, symmetry of mixed partials (statement only), the chain rule, implicit differentiation, exact differentials, and differentiating an integral with respect to a parameter.
3. First order linear equations [2]. Constant coefficients and exponential growth, comparison with discrete equations, modelling such as radioactive decay, and the integrating factor for non-constant coefficients.
4. Nonlinear first order equations [4]. Separable and exact equations, sketching solution curves, equilibria and stability, the logistic equation, chemical kinetics and the logistic map.
5. Higher order linear equations [8]. Complementary function and particular integral, the Wronskian and Abel's theorem, constant coefficients, resonance and damping, step and impulse inputs (Heaviside and Dirac delta), and series solutions.
6. Multivariate functions: applications [5]. Directional derivatives and the gradient, Taylor series in \mathbb{R}^n, classifying extrema with the Hessian, coupled first order systems, phase portraits and stability, and simple PDEs including the wave equation solution f(x+ct) + g(x-ct).

The schedules list no formal prerequisites. In North American terms it mixes the ODE topics from the end of Calculus 2 with partial derivatives and extrema from Calculus 3, plus a first course in differential equations.

Differential Equations topic map

PracticeCalculus does not yet have lessons on second order linear ODEs, coupled systems or phase portraits, so use your example sheets for the higher order block and the lessons above for the rest.

Example problems at the Part IA level

1. An integrating factor. Solve y' + \frac{y}{x} = x for x > 0. The integrating factor is x:

2. Stability of equilibria. For the logistic equation y' = y(1 - y), the equilibria are y = 0 and y = 1. With f(y) = y - y^2 we have f'(y) = 1 - 2y, so

3. A constant coefficient equation. For y'' + 2y' + y = 0, the auxiliary equation \lambda^2 + 2\lambda + 1 = 0 has a repeated root \lambda = -1, so

How to do well in Differential Equations

• Do not skip the calculus review. The first five lectures go fast, and slips in integration by parts or partial fractions cost marks all year.
• Classify before you solve. Decide whether an equation is linear, separable, exact or homogeneous before you start, and the method follows.
• Sketch solution curves. The schedules ask for trajectories and phase portraits, and a quick sketch also catches wrong answers.
• Practise Section I questions under time pressure. They are short and reward fluent technique.

Tip: Analysis I in Lent Term proves many of the results this course uses informally. See the Cambridge Analysis I guide.