8.4. Solving Differential Equations with Exponentials
Although we will solve only linear homogeneous differential equations in this book, matrix exponentials are extremely useful for solving linear nonhomogeneous systems. Say we want to solve the following system using exponentials:
In the previous subsection (Diagonalizing), we found the diagonalized matrix for this system: 
Diagonalized matrices can easily be exponentiated. 
Although finding the eigenvalues and eigenvectors as well as exponentiating takes a bit of work, once we have the matrix in its exponentiated form, we can easily solve for any initial conditions:
For 
and 
we find the following: 
This is equivalent to the answer we obtained in Differential Equations earlier: 
and we find the following: