(Solved): Problem C.31 Determine the Laplace transform of the given Ordinary Differential Equations (ODEs), a ...

Problem C.31 Determine the Laplace transform of the given Ordinary Differential Equations (ODEs), and find the function \( Y(s) \). Do not find the inverse Laplace transform of \( Y(s) \). C.31.a. \( -4 y^{\prime}+9 y=3 e^{2 t}+7 \sinh (5 t), \quad y(0)=0 \) C.31.b. \( y^{\prime \prime}-4 y^{\prime}+3 y=e^{7 t} \cos (4 t), \quad y(0)=2, y^{\prime}(0)=-1 \) C.31.c. \( 2 y^{\prime \prime}+y^{\prime}-2 y=t^{2}+\sin (t-1) u(t-1), \quad y(0)=0, y^{\prime}(0)=1 \) C.31.d. \( -y^{\prime \prime}-2 y^{\prime}-3 y=-\cosh (-4 t) * e^{-5 t}, \quad y(0)=-1, y^{\prime}(0)=0 \)