# Manchester MATH11422 ODEs and Applications: Topics and Practice

> A study guide for MATH11422 ODEs and Applications at Manchester: first and second-order ODEs, phase planes, Euler's method and mechanics, with practice.

Source: https://practicecalculus.com/courses/university-of-manchester-math11422
Updated: 2026-11-08

## At a glance

- **Unit code:** MATH11422
- **Credits:** 20 credits, Level 1
- **Taught:** Year 1, Semester 2
- **Prerequisites:** Not listed on the unit page
- **Assessment:** Written exam 90%, other 10%
- **Covers:** Classifying ODEs, analytical methods for first and second-order ODEs, equilibria and stability, phase planes, Euler's method, Newtonian mechanics, population models
- **Practice here:** [Differential equations in Calculus 2](/learn/calculus-2/improper-integrals-and-applications/slope-fields)

## Key takeaways

- MATH11422 is a 20-credit, Level 1 unit taught in Semester 2 of the Manchester mathematics degrees.
- It teaches analytical, graphical and approximate methods for first and second-order ordinary differential equations.
- It covers equilibrium points, stability and phase plane analysis, plus Euler's method for initial value problems.
- Applications include Newton's laws, the damped driven harmonic oscillator and population models.

**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

| MATH11422 topic | Practice | Read |
| --- | --- | --- |
| What an ODE is, and direction fields | [Slope fields](/learn/calculus-2/improper-integrals-and-applications/slope-fields) | [Differential equations practice](/practice-problems/differential-equations) |
| Separable first-order ODEs | [Separable equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations) | [u-substitution](/blog/u-substitution) |
| Exponential growth and decay | [Growth and decay](/learn/calculus-2/improper-integrals-and-applications/growth-and-decay) | [Exponential functions](/functions/exponential-functions) |
| Population models | [Logistic growth](/learn/calculus-2/improper-integrals-and-applications/logistic-growth) | [Partial fractions](/blog/partial-fractions) |
| Euler's method | [Euler's method](/learn/calculus-2/improper-integrals-and-applications/eulers-method) | [Linear approximation](/blog/linear-approximation) |
| Integration for solving ODEs | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts](/blog/integration-by-parts) |
| Motion, velocity and acceleration | [Second derivative](/learn/calculus-1/applications-of-derivatives/second-derivative) | [Net change](/learn/calculus-1/applications-of-integrals/net-change) |
| Work and energy in mechanics | [Work](/learn/calculus-2/improper-integrals-and-applications/work) | [Antiderivatives](/blog/antiderivatives) |
| Trig and exponential solutions of oscillators | [Trig derivatives](/learn/calculus-1/derivative-rules/trig-derivatives) | [Sine function](/functions/sine-function), [hyperbolic functions](/functions/hyperbolic-functions) |

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

| Unit | Semester | Calculus content |
| --- | --- | --- |
| MATH11121 Mathematical Foundations & Analysis | 1 | Proof, sequences, limits, continuity |
| [MATH11411 Introduction to Vector Calculus](/courses/university-of-manchester-math11411) | 1 | Multivariable and vector calculus |
| [MATH11112 Real Analysis](/courses/university-of-manchester-math11112) | 2 | Series, differentiation and integration, made rigorous |
| MATH11422 ODEs and Applications | 2 | First and second-order differential equations |

## 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

$$y = 3e^{-t^2}$$

**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

$$y = A\cos 2t + B\sin 2t$$

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 $h = 0.1$,

$$y_1 = y_0 + h\,(y_0 + t_0) = 1 + 0.1(1 + 0) = 1.1$$

## How to do well in MATH11422

- **Make integration automatic.** Most first-order ODEs reduce to an integral, so practise [choosing an integration strategy](/learn/calculus-2/integration-techniques/choosing-a-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.

## Frequently asked questions

### What is MATH11422 at Manchester?

MATH11422 ODEs and Applications is a first-year unit that gives a basic introduction to ordinary differential equations. The unit page says it covers analytical, graphical and approximate methods for first and second-order ODEs, with applications from mechanics, population models, economics and biology.

### What topics are in MATH11422?

Classifying ODEs by order, linearity and type of problem; existence and uniqueness; analytical methods for several classes of first and second-order ODEs; equilibrium points and stability of first-order systems; phase plane analysis; Euler's method; Newton's laws and particle motion; the damped, driven harmonic oscillator; and population models.

### How is MATH11422 assessed?

The unit page lists a written exam worth 90% and other assessment worth 10%. There is a supervision each week where your work is marked and discussed.

### What books are recommended for MATH11422?

The unit page recommends Farlow's Introduction to Differential Equations and Their Applications and Braun's Differential Equations and Their Applications, with Trefethen, Birkisson and Driscoll's Exploring ODEs as extra reading.

### How should I prepare for MATH11422?

Make sure integration is fluent, especially substitution, integration by parts and partial fractions, since solving ODEs comes down to integrating. Then practise separable equations and slope fields on PracticeCalculus.