linear ordinary differential equation Find the exact solution of the system: 
$$
\begin{align}
\frac{\operatorname{d}y}{\operatorname{d}t} &= -15y- 25z \\
\frac{\operatorname{d}z}{\operatorname{d}t} &= -47y-85z 
\end{align}
$$
either by the writing the equations in matrix form as $\frac{\operatorname{d}x}{\operatorname{d}t} =Ax$ where $x=\pmatrix{y\\ z}$ and diagonalising the matrix $A$, or otherwise.

Here's the method; I think it is sufficient for you to work from here and find the solution for yourself.
Find the eigenvalues of your matrix $A$ by solving $\text{det}(A-\lambda I)=0$, and then find the corresponding eigenvectors.  As in the 2nd order single system case, you end up with a solution
$$\mathbf{x}=c_1e^{\lambda_1t}\mathbf{a_1}+c_2e^{\lambda_2t}\mathbf{a_2}$$
where $\mathbf{a_i}$ is the eigenvector corresponding to $\lambda_i$
