In your first question, you have . Differentiating this w.r.t. t gives .
Your second equation is . Substituting from the results from manipulating the first equation gives
which is a second order linear constant coefficient ODE. Solve as normal. When you have x, you should be able to find y.
However, because (1a) asks you to find eigenvalues and eigen vectors of the matrix and problem (2b) says "Hence write down the general solution of the second order differential equation" I suspect the student is not intended to change the system of equations to the second order equation but solve the corresponding matrix equation
or "dX/dt= AX"
In problem (1a) we are given that one eigenvalue is 1- 2i and can then find that the other is -1+ 2i. We can then find the corresponding eigenvectors. Letting P be the matrix having those eigenvectors as columns, we have " where D is the diagonal matrix\[tex]\begin{pmatrix}1- 2i & 0 \\ 0 & -1+ 2i\end{pmatrix}
Then we can write the differential equation as dX/dt= AX= P^{-1}DPX or, multiplying on both sides by P, PdX/dt= d(PX)/dt= D(PX) which, letting Z= PX, is dZ/dt= DZ. That now is uncoupled:
is easy to solve:
is easy to solve:
And then, since , .