91Ó°ÊÓ

Prove the following properties of convolution: (a) \(f * g=g * f\). (b) \(f \cdot(g * h)=(f * g) * h .\) (c) \(f *(g+h)=f * g+f * h .\) (d) \(\quad f * 0=0 * f\).

Compute the convolution and its Laplace transform of the following functions: (a) \(e^{t} * e^{2 t}\). (b) \(\sin \omega t * \cos \omega t\). (c) \(t * e^{t}\).

Obtain the solution of each of the Volterra integral equations (a) \(y(t)=t+\int_{0}^{t} \cos (t-\tau) y(\tau) \mathrm{d} \tau\). (b) \(y(t)=e^{t}-\int_{0}^{t}(t-\tau)^{2} y(\tau) \mathrm{d} \tau\).

Solve the following boundary-value problems: (a) \(y^{\prime \prime}+4 y^{\prime}+4 y=t\) \(y(0)=1, \quad y(1)=2 .\) (b) \(y^{\prime \prime}+y=e^{t}\), \(y^{\prime}(0)=0, \quad y(1)=1 .\)

Solve the initial-value problem $$ \begin{array}{c} y^{\prime \prime}-2 y^{\prime}+y=t^{2} \\ y(0)=0, \quad y^{\prime}(0)=0 \end{array} $$ by using a Green's function.

\begin{array}{l} \text { Determine the solution of each of the following initial-value problems: }\\\ \text { (a) } \quad y^{\prime \prime}+4 y=\cos \omega t-\delta(t-1) \text { , }\\\ \begin{array}{l} y(0)=0, \quad y^{\prime}(0)=0 . \\ \text { (b) } \quad y^{\prime \prime \prime}+y^{\prime}=8(t-2)+e^{-t}, \end{array}\\\ y(0)=1, \quad y^{\prime}(0)=0, \quad y^{\prime \prime}(0)=0 . \end{array}

Solve the initial-value problem $$ \begin{aligned} y^{\prime \prime}-y &=f(t) \\ y(0)=0, & y^{\prime}(0)=0, \end{aligned} $$ by (a) the method of the Laplace transform directly, (b) the Laplace transform using a Green's function, (c) the method of variation of parameters.