Darlithoedd

Lecture 18

Summary.

  • Rocket science!

Lecture 17

Summary.

  • Modelling cooling objects

  • Modelling springs

Lecture 16

Summary.

  • Population growth - introducing terms to capture various effects

  • The logistic map

Lecture 15

Summary.

  • Falling objects

  • Population growth

Lecture 14

Summary.

  • Euler-Cauchy equations

  • Example

  • Introduction to modelling

  • Newton's laws of motion

Lecture 13

Summary.

  • Higher order linear ODEs

  • Lagrange's method of variation of parameters

  • Example

Lecture 12

Summary.

  • What to do when the forcing term is the sum or product of different functions

  • Example

Lecture 11

Summary.

  • Modifying the particular integral trial function when the usual trial function is of a form already present in the complementary function

  • Examples

Lecture 10

Summary.

  • Second order inhomogeneous equations - finding the particular integral

  • Method of undetermined coefficients

  • Forcing terms of polynomial, trigonometric, and exponential types

  • Examples

  • An example where it doesn't work!

Lecture 09

Summary.

  • Second order homogeneous equations - finding the complementary function

  • Distinct roots of the auxiliary equation

  • Repeated roots of the auxiliary equation

  • Complex conjugate roots of the auxiliary equation

  • Purely imaginary roots of the auxiliary equation

Lecture 08

Summary.

  • Bernoulli's equation

  • Observations regarding linearity

  • Higher order linear ODEs

Lecture 07

Summary.

  • Examples of integrating factor method

  • Bernoulli's equation

Lecture 06

Summary.

  • \(\frac{dy}{dx}=\frac{ax+by+c}{lx+my+n}\) - what if the usual substitution doesn't work?

  • Linear first order ODEs - \(\frac{dy}{dx}+P(x)y=Q(x)\)

  • Solving them using the integrating factor method

  • Example

Lecture 05

Summary.

  • \(\frac{dy}{dx}=\frac{ax+by+c}{lx+my+n}\)

  • Reduction to homogeneous form

  • Examples

Lecture 04

Summary.

  • First order homogeneous ODEs

  • Using the substitution \(v=\frac{y}{x}\)

  • Examples

Lecture 03

Summary.

  • Separable variables

  • Justification of the abuse of notation of considering \(\frac{\mathrm{d}y}{\mathrm{d}x}\) as a fraction

  • Examples

  • Homogeneous functions of degree \(n\)

  • Examples of such functions

Lecture 02

Summary.

  • Classification practice

  • Direct integration

  • General solutions and particular solutions

  • Observations from examples - an \(n\)th order ODE has at least \(n\) arbitrary constants in its general solution

  • Explicit and implicit solutions

  • Introduction to separable variables

Lecture 01

Summary.

  • How the module works

  • Why study differential equations?

  • Notation

  • What is a differential eqution?

  • Introducing classification: order, degree, and linearity