focused on ODEs. We start off with some basic terminology relating to DEs - linear, homogeneous, 1st and 2nd order etc. Then, we revisit some of the techniques you used in MTH1030 to solve 1st-order ODEs, except in a slightly more rigorous manner. For example, you will probably notice "separable" ODEs are the ones where you could go
or something similar in MTH1030. "Linear" ODEs are the ones where you would use "the integrating factor", except now we cover it a bit more rigorously through understanding the "homogeneous" and "particular" solutions, and the principle of superposition. We also go through two new types of 1st-order ODEs, which are 'exact' and 'homogeneous type' ODEs. These have their own solving methods.
We then look at how we can turn higher-order ODEs into 1st order ODEs, except now with vectors replacing variables. Unfortunately, this does not make them any easier to solve (and most of us were questioning why we would even do such a thing), but it does allow us to more easily state if a solution exists. Remember Euler's method from Specialist Maths? Well, we revisit it, and learn that it is in fact a relatively poor approximation scheme. We look at Heun's method, which is more accurate, and work out a rough guide to "accuracy" of a general class of approximation schemes, which both Euler's and Heun's methods fall under.
Next, we turn to 2nd order ODEs. You probably learnt how to solve ones with constant coefficients in MTH1030; we do that again here. However, we also learn some new techniques for solving 2nd order linear ODEs, with non-constant coefficients, such as the method of "variation of parameters", and the humble "trial solutions" method. We learn how to check if solutions to an ODE are linearly independent (this is important because it is a necessary requirement for many of out solution methods), through the Wronskian matrix. Then, we learn about how to deal with ODEs where we might not have an
present, or a
present, even if
is present (and things like that).
Finally, we learn about series solutions to ODEs. Sometimes, it is very difficult for us to see if a particular ODE has a solution. So what we can try and do is see if an infinite polynomial solution works (much like a Taylor series). There was also a section on Bessel's functions, Legendre's and Frobenius' methods, but we never got around to learning those in lectures.
Throughout this part of the course, there are applications mainly to simple harmonic motion and harmonic oscillators, but also things such as radioactive decay, Newton's Law of Cooling and curves of pursuit are covered. There are also a number of "fundamental theorems", which are basically statements that do not solve an equation, but tell you that a solution exists, or some property of the solutions. You will soon see that it is very important to know the conditions under which these apply, and to invoke them appropriately.