Chapter 4

In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}\left\\{e^{3 t}\left(9-4 t+10 \sin \frac{t}{2}\right)\right\\} $$

Fill in the blanks or answer true/false. $$ \mathscr{L}\left\\{e^{-3 t} \sin 2 t\right\\}= $$____

Find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}\left\\{e^{3 t}\left(9-4 t+10 \sin \frac{t}{2}\right)\right\\} $$

Use the Laplace transform to solve the given differential equation subject to the indicated initial conditions. $$ y^{\prime \prime}+4 y^{\prime}+13 y=\delta(t-\pi)+\delta(t-3 \pi), \quad y(0)=1, y^{\prime}(0)=0 $$

Use the Laplace transform to solve the given system of differential equations. $$ \begin{aligned} &\frac{d^{2} x}{d t^{2}}+3 \frac{d y}{d t}+3 y=0 \\ &\frac{d^{2} x}{d t^{2}}+3 y=t e^{-t} \\ &x(0)=0, x^{\prime}(0)=2, \\ &y(0)=0 \end{aligned} $$

Use the Laplace transform to solve the given initial-value problem. Use the table of Laplace transforms in Appendix III as needed. $$ y^{\prime \prime}+9 y=\cos 3 t, \quad y(0)=2, y^{\prime}(0)=5 $$

In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{1}{(s+2)^{3}}\right\\} $$

Find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{1}{(s+2)^{3}}\right\\} $$

Use the Laplace transform to solve the given system of differential equations. $$ \begin{aligned} &\frac{d x}{d t}=4 x-2 y+2 q(t-1) \\ &\frac{d y}{d t}=3 x-y+q(t-1) \\ &x(0)=0, y(0)=\frac{1}{2} \end{aligned} $$

Use the Laplace transform to solve the given initial-value problem. Use the table of Laplace transforms in Appendix III as needed. $$ y^{\prime \prime}+y=\sin t, \quad y(0)=1, y^{\prime}(0)=-1 $$

In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{1}{(s-1)^{4}}\right\\} $$

Use the Laplace transform to solve the given differential equation subject to the indicated initial conditions. $$ y^{\prime \prime}-7 y^{\prime}+6 y=e^{t}+\delta(t-2)+\delta(t-4), y(0)=0, y^{\prime}(0)=0 $$

Find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{1}{(s-1)^{4}}\right\\} $$

Use the Laplace transform to solve the given initial-value problem. Use the table of Laplace transforms in Appendix III as needed. $$ \begin{gathered} y^{\prime \prime}+16 y=f(t), \quad y(0)=0, y^{\prime}(0)=1, \text { where } \\\ f(t)=\left\\{\begin{array}{lr} \cos 4 t, & 0 \leq t<\pi \\ 0, & t \geq \pi \end{array}\right. \end{gathered} $$

In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{1}{s^{2}-6 s+10}\right\\} $$

Fill in the blanks or answer true/false. $$ \mathscr{L}^{-1}\left\\{\frac{20}{s^{6}}\right\\}= $$____

A uniform beam of length \(L\) carries a concentrated load \(w_{0}\) at \(x=\frac{1}{2} L\). Solve the differential equation $$ E I \frac{d^{4} y}{d x^{4}}=w_{0} \delta\left(x-\frac{1}{2} L\right), 0

Find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{1}{s^{2}-6 s+10}\right\\} $$

Use the Laplace transform to solve the given initial-value problem. Use the table of Laplace transforms in Appendix III as needed. $$ \begin{gathered} y^{\prime \prime}+y=f(t), \quad y(0)=1, y^{\prime}(0)=0, \text { where } \\ f(t)=\left\\{\begin{array}{lr} 1, & 0 \leq t<\pi / 2 \\ \sin t, & t \geq \pi / 2 \end{array}\right. \end{gathered} $$

In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{1}{s^{2}+2 s+5}\right\\} $$

Fill in the blanks or answer true/false. $$ \mathscr{L}^{-1}\left\\{\frac{1}{3 s-1}\right\\}= $$_____

A uniform beam of length \(L\) carries a concentrated load \(w_{0}\) at \(x=\frac{1}{2} L\). Solve the differential equation $$ E I \frac{d^{4} y}{d x^{4}}=w_{0} \delta\left(x-\frac{1}{2} L\right), 0

Find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{1}{s^{2}+2 s+5}\right\\} $$

Someone tells you that the solutions of the two IVPs $$ \begin{array}{ll} y^{\prime \prime}+2 y^{\prime}+10 y=0, & y(0)=0, y^{\prime}(0)=1 \\ \text { and } \quad y^{\prime \prime}+2 y^{\prime}+10 y=\delta(t), \quad y(0)=0, y^{\prime}(0)=0 \end{array} $$ are exactly the same. Do you agree or disagree? Defend your answer.

In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{s}{s^{2}+4 s+5}\right\\} $$

Fill in the blanks or answer true/false. $$ \mathscr{L}^{-1}\left\\{\frac{1}{(s-5)^{3}}\right\\}= $$_____

Find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{s}{s^{2}+4 s+5}\right\\} $$

In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{2 s+5}{s^{2}+6 s+34}\right\\} $$

Find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{2 s+5}{s^{2}+6 s+34}\right\\} $$

In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{s}{(s+1)^{2}}\right\\} $$

Find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{s}{(s+1)^{2}}\right\\} $$

In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{5 s}{(s-2)^{2}}\right\\} $$

Fill in the blanks or answer true/false. $$ \mathscr{L}^{-1}\left\\{\frac{e^{-5 s}}{s^{2}}\right\\}= $$____

In some instances the Laplace transform can be used to solve linear differential equations with variable monomial coefficients. Use Theorem 4.4.1 to reduce the given differential equation to a linear first-order \(\mathrm{DE}\) in the transformed function \(Y(s)=\mathscr{L}\\{y(t)\\}\). Solve the first-order DE for \(Y(s)\) and then find \(y(t)=\mathscr{L}^{-1}\\{Y(s)\\}\). $$ 2 y^{\prime \prime}+t y^{\prime}-2 y=10, \quad y(0)=y^{\prime}(0)=0 $$

Find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{5 s}{(s-2)^{2}}\right\\} $$

In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{2 s-1}{s^{2}(s+1)^{3}}\right\\} $$

Fill in the blanks or answer true/false. $$ \mathscr{L}^{-1}\left\\{\frac{s+\pi}{s^{2}+\pi^{2}} e^{-s}\right\\}= $$____

Find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{2 s-1}{s^{2}(s+1)^{3}}\right\\} $$

In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{(s+1)^{2}}{(s+2)^{4}}\right\\} $$

Fill in the blanks or answer true/false. $$ \mathscr{L}^{-1}\left\\{\frac{1}{L^{2} s^{2}+n^{2} \pi^{2}}\right\\}= $$____

Find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{(s+1)^{2}}{(s+2)^{4}}\right\\} $$

Use Theorem to evaluate the given Laplace transform. Do not evaluate the integral before transforming. $$ \mathscr{L}\left\\{e^{-t} * e^{t} \cos t\right\\} $$

In Problems, use the Laplace transform to solve the given initial-value problem. $$ y^{\prime}+4 y=e^{-4 t}, \quad y(0)=2 $$

Fill in the blanks or answer true/false. $$ \mathscr{L}\left\\{e^{-5 t}\right\\} \text { exists for } s> $$____

Use the Laplace transform to solve the given initial-value problem. $$ y^{\prime}+4 y=e^{-4 t}, \quad y(0)=2 $$

In Problems, use the Laplace transform to solve the given initial-value problem. $$ y^{\prime}-y=1+t e^{t}, \quad y(0)=0 $$

Fill in the blanks or answer true/false. $$ \text { If } \mathscr{L}\\{f(t)\\}=F(s), \text { then } \mathscr{L}\left\\{t e^{8 t} f(t)\right\\}= $$

Use the Laplace transform to solve the given initial-value problem. $$ y^{\prime}-y=1+t e^{t}, \quad y(0)=0 $$

In Problems, use the Laplace transform to solve the given initial-value problem. $$ y^{\prime \prime}+2 y^{\prime}+y=0, \quad y(0)=1, \quad y^{\prime}(0)=1 $$

Fill in the blanks or answer true/false. $$ \text { If } \mathscr{L}\\{f(t)\\}=F(s) \text { and } k>0 \text {, then } \mathscr{L}\left\\{e^{a t} f(t-k) \bullet(t-k)\right\\}= $$____