Solve each of the following problems using the Laplace transform. $y^{\prime \prime}+2 t y^{\prime}-4 y=6 ; y(0)=y^{\prime}(0)=0$

Given

$$f(x)=\left\{\begin{array}{rr} 1, & -2<x<0 \\ -1, & 0<x<2 \end{array}\right.$$

find the exponential Fourier transform $g(\alpha)$ and the sine transform $g_s(\alpha)$. Write f(x) as an integral and use your result to evaluate $\int_{0}^{\infty} \frac{(\cos 2 \alpha-1) \sin 2 \alpha}{\alpha} d \alpha$.

Transform each side of the equation to determine if it is an identity. $(6 g+2 h)^{2}=9 g^{2}+h(6 g+h)$

There is an interesting convolution-type theorem for Fourier sine transforms. Suppose that we want h(x) but know its sine transform $H(\omega)$ to be a product $H(\omega)=\bar{S}(\omega) \bar{C}(\omega),$ where $\bar{S}(\omega)$ is the sine transform of s(x) and $\bar{C}(\omega)$ is the cosine transform of c(x). Assuming that c(x) is even and s(x) is odd, show that $h(x)=\frac{1}{\pi} \int_{0}^{\infty}s(\bar{x})[c(x-\bar{x})-c(x+\bar{x})] d \bar{x}=\frac{1}{\pi} \int_{0}^{\infty} c(\bar{x})[s(x+\bar{x})-s(\bar{x}-x)] d \bar{x}.$

Determine the z-transform for each of the following sequences. Sketch the pole-zero plot and indicate the region of convergence. Indicate whether or not the Fourier transform of the sequence exists. (a)δ[n+5], (b) δ[n-5], (c)

$$(-1)^nu[n]$$

, (d)

$$(½)^{n+1}u[n+3]$$

, (e)

$$(-1/3)^nu[-n-2]$$

, (f)

$$(¼)^nu[3-n]$$

, (g)

$$2^nu[-n]+(¼)^nu[n-1]$$

, (h)

$$(1/3)^{n-2}u[n-2].$$

Describe how to transform the graph of ƒ into the graph of g. Sketch the graphs by hand and support your answer with a grapher.

$$ƒ(x) =0.5^x,g(x) =3*0.5^x+4$$

Describe how to transform the parent function y = x^2 to the graph of each function below. Graph both functions on the same axes. y = -0.25x^2 + 3

Verify the following formula for the transform of the partial derivative $u_{t}.$ $\mathscr{L}\left[u_{t}(x, t)\right]=s U(x, s)-u(x, 0)$

Determine the Laplace transform of the given function f.

$$f ( t ) = e ^ { a ( t - c ) } \cos b ( t - c ) u _ { c } ( t ),$$

where a, b, and c are positive constants.

Find the inverse Laplace transform of the given function

$$\begin{gather*} F(s) = \dfrac{8e^{-7s}}{s^2 + s - 2}. \end{gather*}$$

A cruise ship maintains an average speed of 15 knots in going from San Juan, Puerto Rico, to Barbados, West Indies, a distance of 600 nautical miles.To avoid a tropical storm, the captain heads out of San Juan in a direction of $20^{\circ}$ off a direct heading to Barbados. The captain maintains the 15-knot speed for 10 hours, after which time the path to Barbados becomes clear of storms. Once the turn is made, how long will it be before the ship reaches Barbados if the same 15-knot speed is maintained?

Use the convolution theorem to find the inverse Laplace transform of the given function. \

$$\frac{s+1}{\left(s^2+1\right)^2}$$

Transform the given differential equation to find a nontrivial solution such that $x(0)= 0$.

$$t x ^ { \prime \prime } + 2 ( t - 1 ) x ^ { \prime } - 2 x = 0$$

Use trigonometric identities to transform the left side of the equation into the right side.

$$(1+\sin \theta)(1-\sin \theta)=\cos ^2 \theta$$