In this section we need to take a brief look at systems of differential equations which are larger than 2 x 2. The problem now is not like the first few sections where we looked at nth order differential equations we can't actually come up with a set of formulas which will always work for all systems. Therefore, with this in mind we're going to look at all probable cases for a 3 x 3 system (leaving several details for you for verify at times) and after that a couple of quick comments about 4x4 systems to exemplify how to extend things out to still larger systems and so we'll leave this to you to truly extend things out if you'd like toward.
We will also not be doing any real examples into this section. The point of this section is just to demonstrate how to extend out what we identify about 2 x 2 systems to larger systems.
Firstly the process is identical regardless of the size of the system. Thus, for a system of 3 differential equations along with 3 unknown functions we first place the system in matrix form,
x?' = A x?
Here the coefficient matrix A, is a 3 x 3 matrix. We after that need to find out the eigen-values and eigen-vectors for A and since A is a 3x3 matrix we identify that there will be 3 eigenvalues (containing repeated eigenvalues if there are some).
It is where the process from the 2x2 systems starts to vary. We will require a total of 3 linearly independent solutions to create the general solution. Several of what we know from the 2x2 systems can be brought forward to that point. For illustration, we make out that solutions corresponding to simple eigenvalues (that is they only occur once in the list of eigen-values) will be linearly independent. We identify that solutions from a set of complex conjugate eigen-values will be linearly independent. We also identify how to find a set of linearly independent solutions from a double eigen-value with a single eigenvector.
Here are also a couple of facts regarding to eigenvalues/vectors which we need to review now as well. Initially, provided A has only real entries (that it always will there) all complex eigenvalues will arise in conjugate pairs (that are: l = a + bi) and their related eigenvectors will also be complex conjugates of all. Subsequently, if an eigenvalue has multiplicity k ≥ 2 (that is arises at least twice in the list of eigen-values) so there will be anywhere from 1 to k linearly independent eigenvectors for the eigenvalue.
Along with all these concepts in mind let's start to go through all the possible combinations of eigen-values which we can possibly have for a 3x3 case. Assume that here we also note that for a 3x3 system this is impossible to have only 2 real distinct eigen-values. The only possibilities are to contain 1 or 3 real distinct eigen-values.