Consider the signal

$$x(t)=e^{-5t} u(t)+e^{-βt} u(t)$$

, and denote its Laplace transform by X(s). What are the constructs placed on the real and imaginary parts of β if the region of convergence of X(s) is Re(s)>-3?

Consider the signal

$$x[n]=(1/5)^n, u[n-3]$$

. Evaluate the z-transform of the signal, and specify the corresponding region of convergence.

Sketch or graph the given function (which is assumed to be zero outside the given interval). Represent it using unit step functions. Find its transform. Show the details of your work.

$e^t(0<t<2)$

Solve the following initial value problems by the Laplace transform. (If necessary, use partial fraction expansion. Show all details.)

$y^{\prime \prime}-4 y^{\prime}+4 y=0, \quad y(0)=2.1$, $y^{\prime}(0)=3.9$

Use the linearity of ${L^{-1}}$, partial fraction expansions, and Table (5.3.1) to find the inverse Laplace transform of the function ${\dfrac{s^3 - 2s^2 - 6s - 6}{(s^2 + 2s + 2) \ s^2}}$.

Use the linearity property (7) along with the transforms found in Example 2,

$$\mathcal{L}\left\{e^{a t}\right\}=\frac{1}{s-a}, \quad s>a \quad \text { and } \quad \mathcal{L}\{t\}=\frac{1}{s^2}, \quad s>0,$$

to calculate the Laplace transform $R(s)=\mathcal{L}\{r(t)\}$ of the given function $r(t)$. For what values $s$ does the Laplace transform exist?

$$r(t)=5 e^{-7 t}+t+2 e^{2 t}$$

We derive the multiplication property of the continuous-time Fourier transform. Let x(t) and y(t) be two continuous-time signals with Fourier transforms X(jω) and Y(jω), respectively. Also, let g(t) denote the inverse Fourier transform of 1/2π{X(jω)*Y(jω). (a) Show that

$$f(t)=1/2π∫_{-∞}^{+∞}X(jθ)[1/2π∫_{-∞}^{+∞}Y(j(ω-θ))e^{jωt}dω]dθ.$$

(b) Show that

$$1/2π∫_{∞}^{-∞}Y(j(ω-θ))e^{jωt}dω=e^{jθt}y(t).$$

(c) Combine the results of parts (a) and (b) to conclude that g(t)=x(t)y(t).

When Jefferson assumed the presidency following the election of $1800$ , he expected to transform the national government. Describe President Jefferson's republican vision and his successes and failures in implementing it. Did subsequent Republican presidents advance the same objectives?

Use the given row transformation to transform each matrix. Do not use a calculator.

$$\left[\begin{array}{rr}-1 & 4 \\ 7 & 0\end{array}\right] \begin{array}{r}7 R_1\\\\\end{array}$$

Use the Laplace transform to solve the given differential equation subject to the indicated initial conditions. $y^{\prime \prime}+y=\delta(t-\pi / 2)+\delta(t-3 \pi / 2), \quad y(0)=0, y^{\prime}(0)=0$

The transformation of a figure into its image is described. Describe the transformations that will transform the image back into the original figure. Then write them algebraically. The figure is dilated by a scale factor of 0.5 and translated 6 units left and 3 units up.

Suppose you have used Gaussian elimination to transform the augmented matrix of a linear system into row-echelon form. How can you tell if the system has

(a) exactly one solution?

(b) no solution?

(c) infinitely many solutions?

Solve the following initial value problems by the Laplace transform. (If necessary, use partial fraction expansion. Show all details.)

$y^{\prime \prime}+2 y^{\prime}+2 y=0, \quad y(0)=1$, $y^{\prime}(0)=-3$

Use the linearity of ${L^{-1}}$, partial fraction expansions, and Table (5.3.1) to find the inverse Laplace transform of the function ${2\dfrac{s^3 - 2s^2 + 8}{s^2 \ (s - 2) \ (s + 2)}}$.