It's now time to do solving systems of differential equations. We've noticed that solutions to the system,
x?' = A x?
It will be the form of,
x? = ?h e^{l}^{t}
Here l and ?h are eigenvalues and eigenvectors of the matrix A. We will be working along with 2 x 2 systems therefore it means that we are going to be searching for two solutions, here the determinant of the matrix x?_{1 }(t) and x?_{2} (t).
X = (x?_{1} x?_{2})
They are non-zero.
We are going to start by searching the case where our two eigen-values, l_{1} and l_{2} are real and distinct. Conversely, they will be real, simple eigen-values. Recall suitably that the eigenvectors for easy eigenvalues are linearly independent. It means that the solutions we find from these will also be linearly independent. The matrix X should be nonsingular, herefore these two solutions will be a fundamental set of solutions, if the solutions are linearly independent. The general solution for this case will find be,
x?(t) = c_{1} e^{l}_{1}^{t} ?h^{(1)} + c_{2} e^{l}_{2}^{t} ?h^{(2)}
Remember that each of our illustrations will actually be broken in two illustrations. The first illustration will be solving by the system and the second illustration will be solving by sketching the phase portrait for the system. Phase portraits are not all the time taught in a differential equations course and thus we'll strip those out of the solution process hence if you haven't covered them in your class you can ignore the phase portrait illustration for the system.