Consider the nonhomogeneous system of first-order, linear differential equations \(y_{1}^{\prime}=y_{2}+\cosh t\) and \(y_{2}^{\prime}=y_{1}\) with boundary conditions \(y_{1}(0)=0\) and \(y_{2}(0)=\) \(-\frac{1}{2}\). Solve this initial value problem by the following three methods:

(a) Matrix methods. Use the method of variation of parameters to determine the particular solution.

(b) Laplace transform methods.

(c) Convert the two first-order differential equations into a single second-order differential equation and solve this latter equation.