Solving a Matrix Equation: Decoupling and Eigenvectors

In summary, the conversation is about how to decouple and solve a system of differential equations. The equations are given as \frac{dx}{dt} = 2x+y-t and \frac{dy}{dt}=2x-y+t. The solution involves rewriting the equations as a matrix and finding the eigenvalues and eigenvectors. However, the person asking the question is unsure of how to decouple the equations. The hint given is to differentiate one of the equations with respect to t.
  • #1
Jess89
1
0
Hello, please can someone tell me how to decouple and solve this equation? It was on a problem sheet, but the solution jumped to the decoupled equation... =(

[tex]
\frac{dx}{dt} = 2x+y-t
[/tex]
[tex]
\frac{dy}{dt}=2x-y+t
[/tex]
I know that it can rewritten as
[tex]

\frac{d}{dt}\left[ \begin{array}{cccc} 2 & 1\\ 2& -1 \end{array} \right] \left[\begin{array}{cccc} x\\ y \end{array}\right] + \left[ \begin{array}{cccc} -t\\ t \end{array} \right]

[/tex]

And for that matrix :

[tex]

\left[ \begin{array}{cccc} 2 & 1\\ 2& -1 \end{array} \right]

[/tex]

the eigenvalues and eigenvectors can be worked out.

But I don't know how to decouple =(

Thank you !
 
Last edited:
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  • #2
Welcome to physics forums.

HINT: Try differentiating one of the ODE's with respect to t.

P.S. Since your question is a homework style question, I'm moving it to the homework forums.
 

FAQ: Solving a Matrix Equation: Decoupling and Eigenvectors

What is the purpose of decoupling in a matrix equation?

Decoupling is the process of breaking down a complex matrix equation into simpler equations that can be solved individually. This allows us to solve for the individual components of the matrix separately, making the overall solution more manageable.

How do you decouple a matrix equation?

To decouple a matrix equation, we use techniques such as Gaussian elimination or LU decomposition to eliminate variables and simplify the equation into a triangular form. This allows us to solve for the variables in a step-by-step manner.

What are eigenvectors and why are they important in solving matrix equations?

Eigenvectors are special vectors that do not change direction when multiplied by a matrix. They are important in solving matrix equations because they give us information about the transformation of the matrix and can help us find the solutions to the equation.

How do you find the eigenvectors of a matrix?

To find the eigenvectors of a matrix, we first find the eigenvalues by solving the characteristic equation. Then, for each eigenvalue, we solve the system of equations (A-λI)x=0 to find the corresponding eigenvector. The eigenvectors are then normalized to have a length of 1.

Can a matrix equation have more than one solution?

Yes, a matrix equation can have more than one solution. This can happen when the matrix is not invertible or when there are multiple eigenvectors associated with the same eigenvalue. In such cases, the solutions will form a linear combination of the eigenvectors.

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