Simultaneous differential equations in form $Ax+b$

Question

Given the simultaneous differential equations

$\frac {d^2x}{dt^2} = -2 \frac {dx}{dt} -3y+2$

$\frac {dy}{dt} = \frac {dx}{dt} +2y$

With initial conditions $x(0)=0, \frac {dx}{dt} (0) = 0, y(0)=2$

I have to transform the simultaneous diff equations to $\frac {dx}{dt} = Ax+b$, with $x(0)=c$

Here, $x(t) = \begin{bmatrix} x(t) \\ x'(t) \\ y(t) \end{bmatrix}$

And obtain A, b and c.

In order to write the matrix $A$ I gotta know $x''$, $y'$, which are known, but also $x'$. I know how to continue after that, but that single row of the matrix is what I dont know how to get.

EDIT:

Ok, following @DeepSea answer, I solved the diff eq $y''-y-2=0$, which gives the answer of

$y=ae^t+\frac {b}{e^t} -2$

$y'=ae^t-\frac {b}{e^t}$

Replaced both of them in $x'=y'-2y$ and got $x'=-ae^t -\frac {3b}{e^t} + 4$

But I cant rewrite that as a function of y or y', which are the equations I know the result of.

Answer 1

We have: $x''= y'' - 2y'= -2x'-3y+2= -2(y'-2y)-3y+2=-2y'+y+2\implies y''=y+2\implies y''-y-2=0$. Can you solve for $y$ from this equation. You can infact return to the equation in the form you want in $x', x$ only. Play with these equations to see how you can make it...