Find the solution of each initial value problem using the appropriate initial value Green's function.

a. \(y^{\prime \prime}-3 y^{\prime}+2 y=20 e^{-2 x}, \quad y(0)=0, \quad y^{\prime}(0)=6\).

b. \(y^{\prime \prime}+y=2 \sin 3 x, \quad y(0)=5, \quad y^{\prime}(0)=0\).

c. \(y^{\prime \prime}+y=1+2 \cos x, \quad y(0)=2, \quad y^{\prime}(0)=0\).

d. \(x^{2} y^{\prime \prime}-2 x y^{\prime}+2 y=3 x^{2}-x, \quad y(1)=\pi, \quad y^{\prime}(1)=0\).