Consider the system

\[\begin{array}{r} x^{\prime}=-4 x-y \\ y^{\prime}=x-2 y \end{array}\]

a. Determine the second-order differential equation satisfied by \(x(t)\).

b. Solve the differential equation for \(x(t)\).

c. Using this solution, find \(y(t)\).

d. Verify your solutions for \(x(t)\) and \(y(t)\).

e. Find a particular solution to the system given the initial conditions \(x(0)=1\) and \(y(0)=0\).