(3) is easy enough, just find the eigenvalues and vectors, $\lambda_{1,2},~v_{1,2}$ of the system matrix

$\begin{pmatrix}x(t)\\y(t)\end{pmatrix} = c_1 e^{\lambda_1 t} v_1 + c_2 e^{\lambda_2 t} v_2$

and then further solve for $c_1, c_2$ using the initial conditions.