## UCT MAM1000 lecture notes part 37 – differential equations part vi – second order differential equations

**Second Order differential equations**

We are only going to look at a particular subset of all possible second order differential equations (that is, equations which contain at most second derivatives) but these particular equations are absolutely ubiquitous across every field of science. The particular subset we are going to look at are linear, homogenous second order differential equations with constant coefficients. These can be written in general as:

It is linear because it contains at most (and in this case at least) a single power of in each term. It is homogenous because there is no term which has no powers of (ie. the right hand side is not a constant), and the coefficients and are any real numbers (though you can extend this to having complex numbers very easily). We will see that depending on the relationship between these numbers ( and ) we can have very different behaviour of the equation.…