Chapter 10

Determine \(L^{-1}[F]\). $$F(s)=\frac{2 s+3}{(s+5)^{2}+49}$$.

Solve the given initial-value problem. $$\begin{aligned} &y^{\prime \prime}+y^{\prime}-6 y=30 u_{1}(t) e^{-(t-1)}, \quad y(0)=3\\\ &y^{\prime}(0)=-4 \end{aligned}$$.

Determine \(L^{-1}[F]\). $$F(s)=\frac{4}{s(s+2)^{2}}$$.

Use the Laplace transform to solve the given initial-value problem. $$y^{\prime \prime}+4 y^{\prime}+4 y=\delta(t-4), \quad y(0)=1, \quad y^{\prime}(0)=2$$

Solve the given initial-value problem. $$y^{\prime \prime}+4 y^{\prime}+5 y=5 u_{3}(t), \quad y(0)=2, \quad y^{\prime}(0)=1$$.

Use the Laplace transform to solve the given system of differential equations subject to the given initial conditions. $$\begin{array}{c} \frac{d x_{1}}{d t}=x_{1}+2 x_{2}, \quad \frac{d x_{2}}{d t}=2 x_{1}+x_{2} \\ x_{1}(0)=1, \quad \frac{d x_{1}}{d t}(0)=0 \end{array}$$

Determine \(L^{-1}[F]\). $$F(s)=\frac{2 s+1}{(s-1)^{2}(s+2)}$$.

Solve the given initial-value problem. $$\begin{array}{l} y^{\prime \prime}-2 y^{\prime}+5 y=2 \sin t+u_{\pi / 2}(t)[1-\sin (t-\pi / 2)] \\ y(0)=0, \quad y^{\prime}(0)=0 \end{array}$$.

Use the Laplace transform to solve the given system of differential equations subject to the given initial conditions. $$\begin{aligned} &\frac{d x_{1}}{d t}=2 x_{2}, \quad \frac{d x_{2}}{d t}=-2 x_{1}\\\ &x_{1}(0)=0, \quad x_{2}(0)=1 \end{aligned}$$

Determine \(L^{-1}[F]\). $$F(s)=\frac{2 s+3}{s\left(s^{2}-2 s+5\right)}$$.

Use the Laplace transform to solve the given system of differential equations subject to the given initial conditions. $$\begin{aligned} &\frac{d x_{1}}{d t}=-2 x_{2}, \quad \frac{d x_{2}}{d t}=2 x_{1}+4 x_{2}\\\ &x_{1}(0)=1, \quad x_{2}(0)=1 \end{aligned}$$

Solve the given initial-value problem. $$y^{\prime \prime}-y=8 e^{t}, \quad y(0)=0, \quad y^{\prime}(0)=0$$.

Use the Laplace transform to solve the given system of differential equations subject to the given initial conditions. $$\begin{aligned} &\frac{d x_{1}}{d_{1}}=2 x_{1}+4 x_{2}+16 \sin 2 t_{1}\\\ &\frac{d x_{2}}{d t}=-2 x_{1}-2 x_{2}+16 \cos 2 t\\\ &x_{1}(0)=0, \quad x_{2}(0)=1 \end{aligned}$$

Solve the given initial-value problem. $$y^{\prime \prime}-4 y=12 e^{2 t}, \quad y(0)=2, \quad y^{\prime}(0)=3$$.

Use the Laplace transform to solve the given integral equation. $$x(t)=2 t+\int_{0}^{t} \sin (t-\tau) x(\tau) d \tau$$

Solve the given initial-value problem. $$y^{\prime \prime}-y^{\prime}-2 y=6 e^{-t}, \quad y(0)=0, \quad y^{\prime}(0)=1$$.

Use the Laplace transform to solve the given integral equation. $$x(t)=2 t^{2}+\int_{0}^{t}(t-\tau) x(\tau) d \tau$$

Solve the initial-value problem $$ y^{\prime}-y=f(t), \quad y(0)=1, $$ where $$ f(t)=\left\\{\begin{aligned} 2, & 0 \leq t<1 ,\\\ -1, & t \geq 1. \end{aligned}\right. $$ in the following two ways: (a) Directly using the Laplace transform. (b) Using the technique for solving first-order linear equations developed in Section 1.6.

Solve the given initial-value problem. $$y^{\prime \prime}+y^{\prime}-2 y=3 e^{-2 t}, \quad y(0)=3, \quad y^{\prime}(0)=-1$$.

Use the Laplace transform to solve the given integral equation. $$x(t)=2 t^{2}+\int_{0}^{t} \sin [2(t-\tau)] x(\tau) d \tau$$

The current \(i(t)\) in an RL circuit is governed by the differential equation $$ \frac{d i}{d t}+\frac{R}{L} i=\frac{1}{L} E(t), $$ where \(R\) and \(L\) are constants and \(E(t)\) represents the applied EMF. At \(t=0,\) the switch in the circuit is closed, and the applied EMF increases linearly from \(0 \mathrm{V}\) to \(10 \mathrm{V}\) in a time interval of 5 seconds. The EMF then remains constant for \(t \geq 5 .\) Determine the current in the circuit for \(t \geq 0.\)

Solve the given initial-value problem. $$y^{\prime \prime}-4 y^{\prime}+4 y=6 e^{2 t}, \quad y(0)=1, \quad y^{\prime}(0)=0$$.

Use the Laplace transform to solve the given integral equation. $$x(t)=3+4 \int_{0}^{t} x(t-\tau) \cos \tau d \tau$$

Solve the given initial-value problem. $$y^{\prime \prime}+2 y^{\prime}+y=2 e^{-t}, \quad y(0)=2, \quad y^{\prime}(0)=1$$.

Solve the given initial-value problem. $$y^{\prime \prime}-4 y=2 t e^{t}, \quad y(0)=0, \quad y^{\prime}(0)=0$$.

Solve the given initial-value problem. $$y^{\prime \prime}+3 y^{\prime}+2 y=12 t e^{2 t}, \quad y(0)=0, \quad y^{\prime}(0)=1$$.

Solve the given initial-value problem. $$y^{\prime \prime}+y=5 t e^{-3 t}, \quad y(0)=2, \quad y^{\prime}(0)=0$$.

Solve the given initial-value problem. $$y^{\prime \prime}-y=8 e^{t} \sin 2 t, \quad y(0)=2, \quad y^{\prime}(0)=-2$$.

Solve the given initial-value problem. $$y^{\prime \prime}+2 y^{\prime}-3 y=26 e^{2 t} \cos t, \quad y(0)=1, \quad y^{\prime}(0)=0$$.

Solve the initial-value problem $$ \begin{aligned} x_{1}^{\prime}=2 x_{1}-x_{2}, & x_{2}^{\prime}=x_{1}+2 x_{2} \\ x_{1}(0)=1, & x_{2}(0)=0. \end{aligned} $$

Solve the initial-value problem $$ \begin{aligned} x_{1}^{\prime}=3 x_{1}+2 x_{2}, & x_{2}^{\prime}=-x_{1}+4 x_{2}, \\ x_{1}(0)=-1, & x_{2}(0)=1. \end{aligned} $$