Matrix Exponential: Solve Homework Equation w/ Initial Condition

[tracedinair]

Homework Statement

Given x' = Ax where A =

( 0 1 )
( -1 0 )

Compute the matrix exponential and then find the solution such that x(0) =

( 1 )
( 2 )

Homework Equations

The Attempt at a Solution

I computed the matrix exponential and obtained the matrix,

e^(A) =

( cos(t) sin(t) )
( -sin(t) cos(t) )

But I don't understand how to compute the initial condition. Am I supposed to compute the initial by multiplying the original A by x(0) and then compute the matrix exponential for the new A? Or multiple e^(A) by x(0)? My notes aren't very clear. But those are my only guesses..

Thanks for any help.

 

[Dick]

What you computed was e^(At). And sure, your solution is then x(t)=e^(At)x(0). If you take d/dt of that then you get x'(t)=Ax(t), right?

 

[tracedinair]

Alright, I think I've got it. Compute x(t) then differentiate it?

 

[Dick]

You don't have to differentiate it, I was just pointing out why x(t)=e^(At)x(0) works as a solution. x'(t)=Ax(t) and x(0)=e^(0)x(0). It satisfies the ode and has the right initial condition.

 

[tracedinair]

Alright, thank you.