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Matrix Exponential

  • #1

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.
 
Last edited:

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618
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?
 
  • #3
Alright, I think I've got it. Compute x(t) then differentiate it?
 
  • #4
Dick
Science Advisor
Homework Helper
26,258
618
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.
 
  • #5
Alright, thank you.
 

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