The variables \(x\) and \(y\) are related by the differential equation
\[\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-2 \frac{\mathrm{~d} y}{\mathrm{~d} x}-3 y=4 \mathrm{e}^{-x} .\]
(a) Find the value of the constant \(k\) such that \(y=k x \mathrm{e}^{-x}\) is a particular integral of the differential equation.
(b) Find the solution of the differential equation for which \(y=\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{1}{2}\) when \(x=0\).