Matrix Exponential: Solve Homework Equation w/ Initial Condition

In summary, the conversation discusses computing the matrix exponential for a given linear system and finding the solution using an initial condition. The solution is x(t)=e^(At)x(0), where x(0) is the initial condition and A is the given matrix. Differentiating x(t) shows that it satisfies the given linear system and has the correct initial condition.
  • #1
tracedinair
50
0

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.
 
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  • #2
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
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.
 

What is the Matrix Exponential method?

The Matrix Exponential method is a mathematical technique used to solve systems of linear equations with initial conditions. It involves using the exponential function to find the solution to the system.

What is the importance of initial conditions in using the Matrix Exponential method?

Initial conditions are necessary in order to find a unique solution to the system of linear equations. They provide the necessary starting point for solving the system using the Matrix Exponential method.

How is the Matrix Exponential method different from other methods of solving linear equations?

The Matrix Exponential method is specifically designed for solving systems of linear equations with initial conditions. It involves using matrix operations and the exponential function, making it different from other methods such as Gaussian elimination or Cramer's rule.

What are some real-world applications of the Matrix Exponential method?

The Matrix Exponential method is commonly used in various fields of science and engineering, such as physics, economics, and control systems. It can be applied to model and analyze systems with multiple variables and initial conditions.

Are there any limitations to using the Matrix Exponential method?

The Matrix Exponential method is only applicable to linear systems of equations. It also requires the initial conditions to be known and the matrix to be invertible. Additionally, it may not be the most efficient method for solving large systems of equations.

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