Solving Simple Coupled System Homework

[morenogabr]

Homework Statement

given a general system,
<br /> \frac{df}{dt}=k_{1}g(t)<br />
<br /> \frac{dg}{dt}=-k_{2}f(t)<br />
How could one solve for f_{analytic}. I've used wolfram, so I know what they look like. But how does one begin to solve for them?

Further, how does one find the eigenvalues, eigenmodes and energies for this type of system?

I have been assigned this problem from my research advisor and must admit I do not have much background with coupled systems.

 

[vela]

Let
\mathbf{x} = \begin{pmatrix}f(t)\\g(t)\end{pmatrix}
and express the system of equations in the form \mathbf{x}&#039; = A\mathbf{x}, where A is a 2x2 matrix. That's the matrix you want to diagonalize.