Manchester MATH11422 ODEs and Applications: Topics and Practice

MATH11422 ODEs and Applications is a Level 1 unit at the University of Manchester, taught in Semester 2 and worth 20 credits. It is where first-year mathematicians learn to set up and solve ordinary differential equations, and to see what they say about falling objects, oscillating springs and growing populations. This guide maps each topic to free practice, so you can sharpen the integration and modelling skills the unit depends on.

What MATH11422 covers

According to the official unit page, MATH11422 provides a basic introduction to ordinary differential equations (ODEs) and their applications. It discusses analytical, graphical and approximate methods for general first and second-order ODEs, and applications from Newtonian mechanics, population models, economics and biology.

The stated aims are to:

• Classify ODEs.
• Give analytical methods for first and second-order ODEs.
• Introduce approximate methods (graphical and numerical) for first-order equations and systems.
• Define the main quantities of classical mechanics, such as force, velocity, acceleration and momentum, and discuss Newton's laws of motion and gravity.
• Introduce simple conservative and non-conservative systems of single particles, and model problems that lead to ODEs.

By the end you should be able to classify an ODE (order, linear or nonlinear, autonomous or not, initial or boundary value problem), find analytical solutions for several classes of equations, locate equilibrium points of first-order systems and decide their stability, carry out phase plane analysis, analyse the damped, driven harmonic oscillator, interpret population models, and use Euler's method.

MATH11422 topic map

PracticeCalculus does not yet have lessons on second-order linear ODEs or phase planes, so use the links above to make the underlying calculus automatic and focus your lecture time on those new methods.

Where MATH11422 fits in Year 1

Example problems at the MATH11422 level

1. A separable equation. Solve \frac{dy}{dt} = -2ty with y(0) = 3. Separating, \int \frac{dy}{y} = \int -2t\,dt, so \ln|y| = -t^2 + C and

2. A second-order equation with constant coefficients. For y'' + 4y = 0, try y = e^{mt} to get m^2 + 4 = 0, so m = \pm 2i and

This is simple harmonic motion with angular frequency 2.

3. One step of Euler's method. For y' = y + t with y(0) = 1 and step h = 0.1,

How to do well in MATH11422

• Make integration automatic. Most first-order ODEs reduce to an integral, so practise choosing an integration strategy.
• Classify before you solve. Deciding the order and whether the equation is linear or separable tells you which method to use.
• Sketch the phase line or phase plane. Equilibria and stability are often quicker to see than to compute.
• Use the weekly supervisions. The unit page says work is marked and discussed each week, which is the best place to catch mistakes early.

Tip: always check your solution by substituting it back into the equation. It takes a minute and catches most sign and constant errors.