Solving Simple Coupled System Homework

[morenogabr]

Homework Statement

given a general system,
$$\frac{df}{dt}=k_{1}g(t)$$
$$\frac{dg}{dt}=-k_{2}f(t)$$
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 $\mathbf{x}' = A\mathbf{x}$, where A is a 2x2 matrix. That's the matrix you want to diagonalize.