Conceptual Question: Vector-Matrix Differential Equation

[adamjts]

Hi I'm just having trouble wrapping my head around differential equations with matrices and vectors...
For example:

let y be a vector.
let A(t) be an nxn matrix.

I have the differential equation:
dy/dt = A(t)y

So I think I understand why the solution is

y = ce^A(t)

But I'm having trouble understanding how to actually get information from this. For example, if someone asked be to find the y_i component at some time t, I wouldn't know how to do it. My friend told me to think of the as a taylor expansion, but I'm still not entirely understanding how to do this. Can someone help explain?

 

[andrewkirk]

The Taylor series is
$$\vec y(t)= e^{A(t)}\vec y_0=\sum_{k=0}^\infty \frac{1}{k!}A(t)^k\vec y_0$$
You include only the elements of the infinite sum up to the point at which all components of the most recently added element are smaller than the tolerable error you have decided upon.
So, with that number of elements, the sum is simply a finite sum of vectors, each of which is a finite power of a known matrix $A(t)$, applied to the known vector $\vec y_0$.