91Ó°ÊÓ

Use properties of the Laplace transform and the table of Laplace transforms to determine \(L[f]\). $$f(t)=e^{3 t} \cos 5 t-e^{-t} \sin 2 t$$

Make a sketch of the given function on \([0, \infty)\) and express it in terms of the unit step function. $$f(t)=\left\\{\begin{array}{cc}0, & 0 \leq t<2, \\\3-t, & 2 \leq t<4 ,\\\\-1, & t \geq 4, \end{array}\right.$$

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime \prime}+9 y=7 \sin 4 t+14 \cos 4 t, \quad y(0)=1, \quad y^{\prime}(0)=2\).

Use the Convolution Theorem and the table of Laplace transforms to show that $$ \int_{0}^{x}(x-w)^{a} w^{b} d w=\frac{a ! b !}{(a+b+1) !} x^{a+b+1} $$ \(a > -1, b > -1, x > 0\)

Sketch the given function and determine its Laplace transform. $$f(t)=\left\\{\begin{array}{lr} t, & 0 \leq t \leq 1 \\ 0, & t \geq 1 \end{array}\right.$$

Solve the given Volterra integral equation. $$x(t)=2 e^{3 t}-\int_{0}^{t} e^{2(t-\tau)} x(\tau) d \tau$$

Solve the given Volterra integral equation. $$x(t)=1+2 \int_{0}^{t} \sin (t-\tau) x(\tau) d \tau$$

Solve the given Volterra integral equation. $$x(t)=2\left\\{1+\int_{0}^{t} \cos [2(t-\tau)] x(\tau) d \tau\right\\}$$

Use the Laplace transform to solve the given system of differential equations subject to the given initial conditions. $$\begin{aligned} &\frac{d x_{1}}{d t}=-2 x_{2}, \quad \frac{d x_{2}}{d t}=2 x_{1}+4 x_{2}\\\ &x_{1}(0)=1, \quad x_{2}(0)=1 \end{aligned}$$

Use the Laplace transform to solve the given system of differential equations subject to the given initial conditions. $$\begin{aligned} &\frac{d x_{1}}{d_{1}}=2 x_{1}+4 x_{2}+16 \sin 2 t_{1}\\\ &\frac{d x_{2}}{d t}=-2 x_{1}-2 x_{2}+16 \cos 2 t\\\ &x_{1}(0)=0, \quad x_{2}(0)=1 \end{aligned}$$