Chapter 4

Use the Laplace transform and these inverses to solve the given initial-value problem. $$ y^{\prime \prime}-2 y^{\prime}+5 y=0, \quad y(0)=1, y^{\prime}(0)=3 $$

A series circuit contains an inductor, a resistor, and a capacitor for which \(L=\frac{1}{2} \mathrm{~h}, R=10 \Omega\), and \(C=0.01 \mathrm{f}\), respectively. The voltage $$ E(t)=\left\\{\begin{array}{lr} 10, & 0 \leq t<5 \\ 0, & t \geq 5 \end{array}\right. $$ is applied to the circuit. Determine the instantaneous charge \(q(t)\) on the capacitor for \(t>0\) if \(q(0)=0\) and \(q^{\prime}(0)=0\).

Use the Laplace transform to solve the given integral equation or integrodifferential equation. $$ f(t)=1+t-\frac{8}{3} \int_{0}^{t}(\tau-t)^{3} f(\tau) d \tau $$

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

Use the Laplace transform to solve the given integral equation or integrodifferential equation. $$ t-2 f(t)=\int_{0}^{t}\left(e^{\tau}-e^{-\eta}\right) f(t-\tau) d \tau $$

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

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

Use the Laplace transform to solve the given integral equation or integrodifferential equation. $$ y^{\prime}(t)=1-\sin t-\int_{0}^{t} y(\tau) d \tau, \quad y(0)=0 $$

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

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

Use the Laplace transform to solve the given integral equation or integrodifferential equation. $$ \frac{d y}{d t}+6 y(t)+9 \int_{0}^{t} y(\tau) d \tau=1, \quad y(0)=0 $$

Suppose \(f(t)\) is a function for which \(f^{\prime}(t)\) is piece wise continuous and of exponential order \(c\). Use results in this section and Section \(4.1\) to justify $$ f(0)=\lim _{s \rightarrow \infty} s F(s), $$ where \(F(s)=\mathscr{L}\\{f(t)\\}\). Verify this result with \(f(t)=\cos k t\).

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

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

Solve equation (10) subject to \(i(0)=0\) with \(L, R, C\), and \(E(t)\) as given. Use a graphing utility to graph the solution for \(0 \leq t \leq 3\) $$ \begin{aligned} &L=0.1 \mathrm{~h}, R=3 \Omega, C=0.05 \mathrm{f}\\\ &E(t)=100[q u(t-1)-q u(t-2)] \end{aligned} $$

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

Make up a function \(F(t)\) that is of exponential order, but \(f(t) \quad F^{\prime}(t)\) is not of exponential order. Make up a function \(f(t)\) that is not of exponential order, but whose Laplace transform exists.

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

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

Solve equation (10) subject to \(i(0)=0\) with \(L, R, C\), and \(E(t)\) as given. Use a graphing utility to graph the solution for \(0 \leq t \leq 3\). $$ \begin{aligned} &L=0.005 \mathrm{~h}, R=1 \Omega, C=0.02 \mathrm{f} \\ &E(t)=100[t-(t-1) ?(t-1)] \end{aligned} $$

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

Suppose that \(\mathscr{L}\left\\{f_{1}(t)\right\\} \quad F_{1}(s)\) for \(s>c_{1}\) and that \(\mathscr{L}\left\\{f_{2}(t)\right\\} \quad F_{2}(s)\) for \(s>c_{2}\). When does \(\mathscr{L}\left\\{f_{1}(t)+f_{2}(t)\right\\}\) \(F_{1}(s)+F_{2}(s) ?\)

The Laplace transform \(\mathscr{L}\left\\{e^{-r^{2}}\right\\}\) exists, but without finding it solve the initial-value problem $$ y^{\prime \prime}+9 y=3 e^{-t^{2}}, y(0)=0, y^{\prime}(0)=0 $$

Solve the integral equation $$ f(t)=e^{t}+e^{t} \int_{0}^{t} e^{-\tau} f(\tau) d \tau $$

The function \(f(t) \quad 2 t e^{t^{2}} \cos e^{t^{2}}\) is not of exponential order. Nevertheless, show that the Laplace transform \(\mathscr{L}\left\\{2 t e^{t^{2}} \cos e^{t^{2}}\right\\}\) exists. [Hint: Use integration by parts.]

If \(\mathscr{L}\\{f(t)\\} \quad F(s)\) and \(a>0\) is a constant, show that $$ \mathscr{L}\\{f(a t)\\}=\frac{1}{a} F\left(\frac{s}{a}\right) $$ This result is known as the change of scale theorem.

In Problems, write each function in terms of unit step functions. Find the Laplace transform of the given function. $$ f(t)=\left\\{\begin{array}{lr} 2, & 0 \leq t<3 \\ -2, & t \geq 3 \end{array}\right. $$

Write each function in terms of unit step functions. Find the Laplace transform of the given function. $$ f(t)=\left\\{\begin{array}{lr} 2, & 0 \leq t<3 \\ -2, & t \geq 3 \end{array}\right. $$

In Problems, write each function in terms of unit step functions. Find the Laplace transform of the given function. $$ f(t)=\left\\{\begin{array}{lr} 1, & 0 \leq t<4 \\ 0, & 4 \leq t<5 \\ 1, & t \geq 5 \end{array}\right. $$

Write each function in terms of unit step functions. Find the Laplace transform of the given function. $$ f(t)=\left\\{\begin{array}{lr} 1, & 0 \leq t<4 \\ 0, & 4 \leq t<5 \\ 1, & t \geq 5 \end{array}\right. $$

In Problems, write each function in terms of unit step functions. Find the Laplace transform of the given function. $$ f(t)=\left\\{\begin{array}{lr} 0, & 0 \leq t<1 \\ t^{2}, & t \geq 1 \end{array}\right. $$

Write each function in terms of unit step functions. Find the Laplace transform of the given function. $$ f(t)=\left\\{\begin{array}{rr} 0, & 0 \leq t<1 \\ t^{2}, & t \geq 1 \end{array}\right. $$

In Problems, write each function in terms of unit step functions. Find the Laplace transform of the given function. $$ f(t)=\left\\{\begin{array}{lr} 0, & 0 \leq t<3 \pi / 2 \\ \sin t, & t \geq 3 \pi / 2 \end{array}\right. $$

Write each function in terms of unit step functions. Find the Laplace transform of the given function. $$ f(t)=\left\\{\begin{array}{lr} 0, & 0 \leq t<3 \pi / 2 \\ \sin t, & t \geq 3 \pi / 2 \end{array}\right. $$

In Problems, write each function in terms of unit step functions. Find the Laplace transform of the given function. $$ f(t)=\left\\{\begin{array}{lr} t, & 0 \leq t<2 \\ 0, & t \geq 2 \end{array}\right. $$

Write each function in terms of unit step functions. Find the Laplace transform of the given function. $$ f(t)=\left\\{\begin{array}{rr} t, & 0 \leq t<2 \\ 0, & t \geq 2 \end{array}\right. $$

In Problems, write each function in terms of unit step functions. Find the Laplace transform of the given function. $$ f(t)=\left\\{\begin{array}{lr} \sin t, & 0 \leq t<2 \pi \\ 0, & t \geq 2 \pi \end{array}\right. $$

Write each function in terms of unit step functions. Find the Laplace transform of the given function. $$ f(t)=\left\\{\begin{array}{lr} \sin t, & 0 \leq t<2 \pi \\ 0, & t \geq 2 \pi \end{array}\right. $$

Show how to use the Laplace transform to find the numerical value of the improper integral \(\int^{\infty} t e^{-2 t} \sin 4 t d t\).

In Problems, use the Laplace transform to solve the given initial-value problem. $$ y^{\prime}+y=f(t), \quad y(0)=0, \text { where } f(t)=\left\\{\begin{array}{lr} 0, & 0 \leq t<1 \\ 5, & t \geq 1 \end{array}\right. $$

Use the Laplace transform to solve the given initial-value problem. $$ y^{\prime}+y=f(t), \quad y(0)=0, \text { where } f(t)=\left\\{\begin{array}{rr} 0, & 0 \leq t<1 \\ 5, & t \geq 1 \end{array}\right. $$

In Problems, use the Laplace transform to solve the given initial-value problem. $$ y^{\prime}+y=f(t), \quad y(0)=0, \text { where } f(t)=\left\\{\begin{array}{lr} 1, & 0 \leq t<1 \\ -1, & t \geq 1 \end{array}\right. $$

Use the Laplace transform to solve the given initial-value problem. $$ y^{\prime}+y=f(t), \quad y(0)=0, \text { where } f(t)=\left\\{\begin{array}{lr} 1, & 0 \leq t<1 \\ -1, & t \geq 1 \end{array}\right. $$

In Problems, use the Laplace transform to solve the given initial-value problem. $$ y^{\prime}+2 y=f(t), \quad y(0)=0, \text { where } f(t)=\left\\{\begin{array}{lr} t, & 0 \leq t<1 \\ 0, & t \geq 1 \end{array}\right. $$

Use the Laplace transform to solve the given initial-value problem. $$ y^{\prime}+2 y=f(t), \quad y(0)=0, \text { where } f(t)=\left\\{\begin{array}{rr} t, & 0 \leq t<1 \\ 0, & t \geq 1 \end{array}\right. $$

In Problems, use the Laplace transform to solve the given initial-value problem. $$ \begin{gathered} y^{\prime \prime}+4 y=f(t), \quad y(0)=0, y^{\prime}(0)=-1, \text { where } \\\ f(t)=\left\\{\begin{array}{lr} 1, & 0 \leq t<1 \\ 0, & t \geq 1 \end{array}\right. \end{gathered} $$

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime \prime}+4 y=f(t), \quad y(0)=0, y^{\prime}(0)=-1\), where $$ f(t)=\left\\{\begin{array}{rr} 1, & 0 \leq t<1 \\ 0, & t \geq 1 \end{array}\right. $$

In Problems, use the Laplace transform to solve the given initial-value problem. $$ y^{\prime \prime}-5 y^{\prime}+6 y=\mathcal{u}(t-1), \quad y(0)=0, y^{\prime}(0)=1 $$

Use the Laplace transform to solve the given initial-value problem. $$ y^{\prime \prime}-5 y^{\prime}+6 y=9(t-1), \quad y(0)=0, \quad y^{\prime}(0)=1 $$

In Problems, use the Laplace transform to solve the given initial-value problem. $$ \begin{gathered} y^{\prime \prime}+y=f(t), \quad y(0)=0, y^{\prime}(0)=1, \text { where } \\ f(t)=\left\\{\begin{array}{ll} 0, & 0 \leq t<\pi \\ 1, & \pi \leq t<2 \pi \\ 0, & t \geq 2 \pi \end{array}\right. \end{gathered} $$