91Ó°ÊÓ

Find the Laplace transforms of the following functions. Show the details of your work. \((a, b, k, \omega, \theta\) are constants.) $$e^{2 a-2 b t}$$

Showing the details, find, graph, and discuss the solution. $$\begin{aligned}&y^{\prime \prime}+2 y^{\prime}+5 y=25 t-1008(t-\pi)\\\&y(0)=-2, \quad y^{\prime}(0)=5\end{aligned}$$

Showing the details of your work, find \(\mathscr{L}(f)\) if \(f(t)\) equals: $$t^{2} \sin \omega t$$

Showing the details of your work, find \(\mathscr{L}(f)\) if \(f(t)\) equals: $$t \cos \omega t$$

Solve the following initial value problems by the Laplace transform. (If necessary, use partial fraction expansion as in Example \(4 .\) Show all details.) $$y^{\prime}+4 y=0, \quad y(0)=2.8$$

Find the Laplace transforms of the following functions. Show the details of your work. \((a, b, k, \omega, \theta\) are constants.) $$-8 \sin 0.2 t$$

Using the Laplece transform and showing the details of your work, solve the initial value problem: $$y_{1}^{\prime}=y_{2}+1-u(t-1)$$$$y_{2}^{\prime}=-y_{1}+1-u(t-1), \quad y_{1}(0)=0$$$$y_{2}(0)=0$$

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. $$20 \cos \pi t(3 < t < 6)$$

Solve the following initial value problems by the Laplace transform. (If necessary, use partial fraction expansion as in Example \(4 .\) Show all details.) $$y^{\prime}+\frac{1}{2} y=17 \sin 2 t, \quad y(0)=-1$$

Find \(f(x)\) if \(\mathscr{S}(f)\) equals.$$\frac{1}{s\left(s^{2}+4\right)}$$