Solving a Matrix Equation: Decoupling and Eigenvectors

[Jess89]

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... =(

$$\frac{dx}{dt} = 2x+y-t$$
$$\frac{dy}{dt}=2x-y+t$$
I know that it can rewritten as
$$<br /> \frac{d}{dt}\left[ \begin{array}{cccc} 2 & 1\\ 2& -1 \end{array} \right] \left[\begin{array}{cccc} x\\ y \end{array}\right] + \left[ \begin{array}{cccc} -t\\ t \end{array} \right]<br />$$

And for that matrix :

$$<br /> \left[ \begin{array}{cccc} 2 & 1\\ 2& -1 \end{array} \right] <br />$$

the eigenvalues and eigenvectors can be worked out.

But I don't know how to decouple =(

Thank you !

 

[Hootenanny]

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.