Since many of you don't have a background in linear algebra, we'll concentrate on systems. We spent a lot of effort solving second-order linear equations, so it's useful to know that these systems can be solved by reducing them to second-order equations.
For example, consider the system
In general, with a system
Here's an example, this time homogeneous:
The one remaining case is where and . But this is actually the easiest case, where the two equations are ``uncoupled'', one only involving and the other only involving . We can solve these two first-order linear equations separately. For example:
Note that we use different names for the arbitrary constants in the
two equations: