Linear Algebra with Differential Equations/Heterogeneous Linear Differential Equations/Diagonalization

First of all (and kind of obvious suggested by the title), must be diagonalizable. Second, the eigenvalues and eigenvectors of are found, and form the matrix which is an augemented matrix of eigenvectors, and which is a matrix consisting of the corresponding eigenvalues on the main diagonal in the same column as their corresponding eigenvectors. Then with our central problem:

We substitute:

Then left multiply by

As a consequence of Linear Algebra we take the following identity:


And because of the nature of the diagonal the problem is a series of one-dimensional normal differential equations which can be solved for and used to find out .