Aims of the project

You are invited to study the solutions of a list of ordinary differential equations using whatever methods you have at your disposal.

Mathematical ideas used

Some of the equations are capable of analytic solution using such mathematical tools as: separation of variables, integrating factors or series solutions. Others are examples of homogeneous, or constant coefficient differential equations. What you can bring to bear will very much depend on your mathematical background at this point.

MATLAB techniques used

All the numerical techniques required have been introduced in Chapter 7. For example, grain (or phase) plot analysis and the numerical solution of coupled first order equations. You may find helpful the M-files associated with that work (fodesol.m, species.m, vderpol.m,…). You can use these directly or copy and modify them as required.

Strategy

Your aim, for each equation in the list of exercises, is to provide the following information as appropriate:

1. (a) For first order equations, a grain plot with typical solutions superimposed. For second order, or coupled first order equations, sketch of a typical phase plot.

2. (b) An analytic general solution if yon can find one.

3. (c) The particular solution for the specified initial conditions.

4. (d) Any other comments you wish to make on the nature of the solutions, their stability etc.

In each case, start by classifying the type of the differential equation. Is it linear? What order is it? Has it got constant coefficients? Is it homogeneous? If it is of a type which you recognise, then try to solve it ‘analytically’, that is, by paper and pencil.