I am asked to find the eigenvalues and eigenvectors of the matrix: $\displaystyle A = \left(\begin{array}{cc}5&3\\1&7\end{array}\right)$

I find them as $\displaystyle \lambda_1 = 8 $, $\displaystyle \lambda_2 = 4 $ and $\displaystyle v_1 = (1,1) $, $\displaystyle v_2 = (-3,1) $

The equations I need to solve are: $\displaystyle \frac{dx}{dt} = 5x +3 y $ and $\displaystyle \frac{dy}{dt} = x + 7y$. These equations are coupled (whatever that means...)

I turn these into a pair of simultaneous equations sort of thing to form the matrix equation $\displaystyle X' = AX $, where X' is $\displaystyle \left(\begin{array}{cc}\frac{dx}{dt}\\\frac{dy}{dt }\end{array}\right)$ and $\displaystyle X = \left(\begin{array}{cc}x&\\y&\end{array}\right)$

As far as I know, I have diagonalise A which will give me $\displaystyle A = P^{-1}DP$. I know $\displaystyle P = \left(\begin{array}{cc}1&-3\\1&1\end{array}\right)$ and $\displaystyle D = \left(\begin{array}{cc}8&0\\0&4\end{array}\right)$ using the eigenvalues.

So I go along...

$\displaystyle X' = P^{-1}DPX $

$\displaystyle PX' = DPX $

$\displaystyle (PX)' = D(PX) $

Subbing matrices in...

$\displaystyle \left(\begin{array}{cc}1&-3\\1&1\end{array}\right) * \left(\begin{array}{cc}\frac{dx}{dt}\\\frac{dy}{dt }\end{array}\right) = \left(\begin{array}{cc}8&0\\0&4\end{array}\right)* \left(\begin{array}{cc}1&-3\\1&1\end{array}\right)*\left(\begin{array}{cc}x& \\y&\end{array}\right)$

Can someone check if what I've done is correct so far? And if so, am I allowed to expend all the differentials and all the matrices then treat them as simultaneous equations. Also, I'm not too sure how to get rid of the $\displaystyle dt$ when there is no $\displaystyle t$ variable in the equations...