Use the Laplace transform to solve the initial value problem. $3 x^{\prime}-y=2 t, x^{\prime}+y^{\prime}-y=0 ; x(0)=y(0)=0$

Use the definition of the Laplace transform, to find $\mathcal{L}\{f(t)\}$ for each of the given functions f(t). $f(t)=e^{-a t} \cos b t$

Use the Laplace transform to solve the first-order initial value problems. $y^{\prime}-2 y=e^{-t} \cos t$, y(0)=-2

Determine the Laplace transform of the given function.

$$f ( t ) = \cos t , \quad 0 \leq t < \pi , \quad f ( t + \pi ) = f ( t ).$$

Find the type, transform to normal form, and solve. Show your work in detail.

$$U_{xx}+2u_{xy}+u_{yy}=0$$

describe how to transform the graph of y=ln x into the graph of the given function. Sketch the graph by hand and support your sketch with a grapher.

$$f(x)=ln(-x)-2$$

Use the Laplace transform to solve the given initial-value problem.

$$y ^ { \prime } + y = 5 e ^ { t } \sin t , \quad y ( 0 ) = 1.$$

Determine the trigonometric interpolating polynomial of degree 4 for $f(x)=x(\pi-x)$ on the interval $[-\pi, \pi]$ using: a. Direct calculation; b. The Fast Fourier Transform Algorithm.

Use the Laplace transform to solve the given initial-value problem. $2 \frac{d y}{d t}+y=0, \quad y(0)=-3$

Use trigonometric identities to transform one side of the equation into the other $(0<\theta<\pi / 2)$. $\tan \theta \cot \theta=1$

Use properties of the Laplace transform and the table of Laplace transforms to determine L[f].

$$f ( t ) = \int _ { 0 } ^ { t } ( t - w ) \cos 2 w d w.$$

Make up two functions $f_{1} \text { and } f_{2}$ that have the same Laplace transform. Do not think profound thoughts.

Determine the inverse Laplace transform of the given function.

$$F ( s ) = \frac { s - 2 } { ( s + 1 ) \left( s ^ { 2 } + 4 \right) }.$$

Find the type, transform to normal form, and solve. Show your work in detail.

$$u_{xx}+2u_{xy}+10u_{yy}=0$$