91Ó°ÊÓ

Sketch on the same axes graphs of \(\sin t, \sin (t-\pi / 2)\), and \(\sin (t+\pi / 2)\), and observe which way the graph shifts.

By using Laplace transforms, solve the following differential equations subject to the given initial conditions. $$ y^{\prime \prime}-2 y^{\prime}+y=2 \cos t, \quad y_{0}=5, \quad y_{0}^{\prime}=-2 $$

By using Laplace transforms, solve the following differential equations subject to the given initial conditions. $$ y^{\prime \prime}+4 y^{\prime}+5 y=2 e^{-2 t} \cos t, \quad y_{0}=0, \quad y_{0}^{\prime}=3 $$

(a) Represent as an exponential Fourier transform the function $$ f(x)=\left\\{\begin{array}{cl} \sin x, & 0

By using Laplace transforms, solve the following differential equations subject to the given initial conditions. $$ y^{\prime \prime}+2 y^{\prime}+10 y=-6 e^{-t} \sin 3 t, \quad y_{0}=0, \quad y_{0}^{\prime}=1 $$

Solve the following sets of equations by the Laplace transform method. $$ \begin{array}{ll} y^{\prime}+z=2 \cos t & y_{0}=-1 \\ z^{\prime}-y=1 & z_{0}=1 \end{array} $$

Solve the following sets of equations by the Laplace transform method. $$ \begin{aligned} &y^{\prime}+z^{\prime}-2 y=1 \quad y_{0}=z_{0}=1 \\ &z-y^{\prime}=t \end{aligned} $$

Solve the following sets of equations by the Laplace transform method. $$ \begin{array}{ll} y^{\prime}+2 z=1 & y_{0}=0 \\ 2 y-z^{\prime}=2 t & z_{0}=1 \end{array} $$

Solve the following sets of equations by the Laplace transform method. $$ \begin{array}{ll} y^{\prime}+2 z=1 & y_{0}=0 \\ 2 y-z^{\prime}=2 t & z_{0}=1 \end{array} $$

Solve the following sets of equations by the Laplace transform method. $$ \begin{aligned} &z^{\prime}+2 y=0 & y_{0}=z_{0}=0 \\ &y^{\prime}-2 z=2 \end{aligned} $$