$x'' = 2 x' +6y +3$
$y' = -x' -2y$
subject the the initial condition
$x(0) = 0; x'(0) = 0; y(0) = 1$
The first part of the question is about finding $e^{At}$ of this matrix $A = \begin{bmatrix} 0 & 1& 0 \\[0.3em] 0 & 2 & 5 \\[0.3em] 0 & -1 & -2 \end{bmatrix}$ and I already found $e^{At}$.
I observed that entries of the matrix is similar to the question but not sure how it works. How to proceed after finding $e^{At}$?
we can look for a particular solution of the form $$x = a, x' = b, y = c \to 0 = 2b + 6c + 3, \, 0 = -b - 2c \to a = 0,b = 3, c = -\frac 32 $$
now, make a change of variable $u = x, v = x'-3, w = y + \frac 32.$ so that $$\pmatrix{u'&v'&w'}^T= A\pmatrix{u&v&w}^T, u = 0, v = -3, w = \frac 52 \text{ at } t = 0 \\\to (u,v,w)^T = e^{At}(0,-3,\frac 52)^T, (x,x',y) = (0,3, -\frac 52)^T + e^{At}(0,-3,\frac 52)^T.$$