All the content you will need for 2nd year Differential Equations

Laplace's equation for a steady state equation. Steady State heat equation finding a fourier series solution. Computer algebra to find integrals. Matlab script for graphing fourier solutions. Solving Laplace's equation with Dirichlet and/or Neumann boundary conditions with an example.

D'Alembert's solution and derivation of the form of solutions. Classification of PDEs with an example using the wave equation.

Linearising wave equation for a water column. Using separation of variables to create a general solution to the wave equation.

Matlab Script graphing Wave Equation solution of separation of variables with Fourier series. Heat Flow equation example solving and graphing with Matlab. PDEs and waves, shows part way through deriving the PDE for the wave equation.

Graph and solve fourier series for square wave with arbitrary period. Fourier Cosine and Sine series with matlab example. Half-range expansion with examples.

Theorem for Fourier series, the fourier series and the integrals to solve the coefficients. Forced oscillation example and graphing of solution. Generalising Fourier series to an Arbitrary period and derivation.

Separation of variables to solve the Heat equation and represent as a summation solution. Example of copper bar and solving for its solution to transfer of heat over time. Includes matlab code of graphing of solution. Different Boundary conditions: Dirichlet and Neumann boundary conditions, motivati...

Square Wave example with matlab script to plot solution. Theorem of functions being orthogonal and proving this is the case for different periods. Deriving the value of the coefficients for the fourier series. Square wave Redux.

Example of Near constant car density continued with interpretation. D'Alembert's solutions with an example of some cars being able to pass other cars. Heat equation and solving it with separation of variables coming to the full general solution.

Partial Differential Equations and its definition and properties. Some important PDEs are given and an example given of how to solve one. It goes through how to make a well-posed PDE question and the modelling framework. An example is given of modelling car traffic going through the steps of the mod...

Solving systems of equations using matlab and creating a general solution. Shows example of a repeated root problem and how to solve. Goes through the theorem for system of equations and a proof. Gives an example of modelling a system of equations.

Looking at steady state solution for modelling the populations of predators and prey with graphs. Constructing systems of equations for different stages of life to model the population of a species.

Reduction of order method to find all terms of the homogeneous solution. Euler Cauchy Equations and the types of solution for the roots of the characteristic equation. Example using Reduction of order finding the general solution. Modelling Mathematical ecology, by looking at a fish example, finding...

Damped system and the different scenarios and their solutions. Solutions plotted in matlab and graphed. Looked at Resonance in forced oscillations and the particular solution for this equation. Graph the solution of types of solution for the particular solution for different constant values and inte...

General solution for two nonhomogeneous ODE example. How to use matlab to check your solutions. Modelling free and forced oscillations theory and finding the solutions to an undamped system. Example of a harmonic oscillator and looking at the different types of damped systems.

Proof for multiple roots for solutions. Theory for the general solution for nonhomogeneous linear ODEs and an example. Method of undetermined guessing and an example which is plotted with matlab code. An example of an IVP for nonhomogeneous solutions.

Existence and completeness theorem and a general proof (not complete), Characteristic Equation derivation and the different types with an example. Theory of linear independent solutions and a basis, using Wronksian for linear independence and basis- partial proof. Complex roots theory and an example...

Picard's Iteration method with example and using matlab to solve. Classification of ODEs: order and linearity. Initial value problem theory: autonomy, deterministism, homogeneous linear ODEs and superposition with proof. Different cases of equation giving examples of the different types and features...

Example finished from the end of the last notes. Definitions of linearly independent, linear dependence and linear combination with an example. Existence and uniqueness theorem. Example of solving a third order IVP. Definition of the Wronskian, the Wronskian test with examples and a proof.

Singular solutions for ODEs with examples and a graph. Existence and uniqueness theorem example. Picard's Iteration Method theory and an example.