I would really appreciate it if someone could show me the steps to how to solve such a differential equation.
Thank you very much.
There are at least a couple of options:
1. Use the eigenvalue/eigenvector diagonalization approach.
2. Eliminate one variable to get a second-order DE.
You can check those out at Chris's DE tutorial. Is a method specified for this problem, or can you use any method?
Now we have
Here is a link on how to solve a matrix system ODE. Scroll down a bit to find the computation of eigenvalues and eigenvectors section. They also have examples of this type of problem.
Eigenvalues and Eigenvectors Technique