Chapter 10

Make a sketch of the given function on \([0, \infty)\) and express it in terms of the unit step function. $$f(t)=\left\\{\begin{array}{cr}1, & 0 \leq t<\pi / 2, \\\\\sin t, & \pi / 2 \leq t<3 \pi / 2, \\ -1, & t \geq 3 \pi / 2.\end{array}\right.$$

Use the linearity of \(L\) and the formulas derived in this section to determine \(L[f]\). $$f(t)=2 t-e^{2 t}$$

The motion of a spring-mass system is governed by $$\begin{array}{c} \frac{d^{2} y}{d t^{2}}+4 \frac{d y}{d t}+13 y=10 \sin 5 t \\ y(0)=0, \quad \frac{d y}{d t}(0)=0 \end{array}$$ At \(t=10\) seconds, the mass is dealt a blow in the downward (positive) direction that instantaneously imparts 2 units of impulse to the system. Determine the resulting motion of the mass.

Determine the inverse Laplace transform of \(F.\) $$F(s)=\frac{s e^{-s}}{s^{2}+4}$$.

Make a sketch of the given function on \([0, \infty)\) and express it in terms of the unit step function. $$f(t)=\left\\{\begin{array}{rr}\sin t, & 2 n \pi \leq t<(2 n+1) \pi \\\& (n=0,1,2,3, \ldots) \\ 0, & \text { otherwise }\end{array}\right.$$

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime \prime}-y=12 e^{2 t}, \quad y(0)=1, \quad y^{\prime}(0)=1\).

Use the linearity of \(L\) and the formulas derived in this section to determine \(L[f]\). \(f(t)=\sinh b t,\) where \(b\) is constant.

Determine the inverse Laplace transform of the given function. $$F(s)=\frac{4}{s^{2}}-\frac{s+2}{s^{2}+9}.$$

use properties of the Laplace transform and the table of Laplace transforms to determine \(L[f]\) $$f(t)=e^{-5 t} / \sqrt{t}$$

Determine \(L^{-1}[F(s) G(s)]\) in the following two ways: (a) using the Convolution Theorem, (b) using partial fractions. $$F(s)=\frac{1}{s}, \quad G(s)=\frac{1}{s-2}$$

Determine \(L^{-1}[F(s) G(s)]\) in the following two ways: (a) using the Convolution Theorem, (b) using partial fractions. $$F(s)=\frac{1}{s+1}, \quad G(s)=\frac{1}{s}$$

Determine \(f(t)\). $$f(t-3)=t e^{-(t-3)}$$.

Determine the inverse Laplace transform of \(F.\) $$F(s)=\frac{e^{-3 s}}{s^{2}+1}$$.

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime \prime}+4 y=10 e^{-t}, \quad y(0)=4, \quad y^{\prime}(0)=0\).

Use the linearity of \(L\) and the formulas derived in this section to determine \(L[f]\). \(f(t)=\cosh b t,\) where \(b\) is constant.

Determine the inverse Laplace transform of the given function. $$F(s)=\frac{2}{s}-\frac{3}{s+1}.$$

Use properties of the Laplace transform and the table of Laplace transforms to determine \(L[f]\). $$f(t)=2(t-5) u_{5}(t)$$

Consider the spring-mass system whose motion is governed by the initial-value problem $$\begin{aligned} \frac{d^{2} y}{d t^{2}}+\omega_{0}^{2} y &=F_{0} \sin \omega t+A \delta\left(t-t_{0}\right) \\ y(0) &=0, \quad \frac{d y}{d t}(0)=0 \end{aligned}$$ where \(\omega_{0}, \omega, F_{0}, A,\) and \(t_{0}\) are positive constants and \(\omega \neq \omega_{0} .\) Solve the initial-value problem to determine the position of the mass at time \(t\)

Determine \(L^{-1}[F(s) G(s)]\) in the following two ways: (a) using the Convolution Theorem, (b) using partial fractions. $$F(s)=\frac{s}{s^{2}+4}, \quad G(s)=\frac{2}{s}$$

Determine the inverse Laplace transform of \(F.\) $$F(s)=\frac{e^{-2 s}}{s+2}$$.

Determine the inverse Laplace transform of the given function. $$F(s)=\frac{s-2}{(s+1)\left(s^{2}+4\right)}.$$

Use properties of the Laplace transform and the table of Laplace transforms to determine \(L[f]\). $$f(t)=2+2\left(e^{-t}-1\right) u_{1}(t)$$

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime \prime}-y^{\prime}-6 y=6\left(2-e^{t}\right), \quad y(0)=5, \quad y^{\prime}(0)=-3\).

Use the linearity of \(L\) and the formulas derived in this section to determine \(L[f]\). $$f(t)=7 e^{-2 t}+1$$

Use properties of the Laplace transform and the table of Laplace transforms to determine \(L[f]\). $$f(t)=\int_{0}^{t}(t-w) \cos 2 w d w$$

Determine \(L^{-1}[F(s) G(s)]\) in the following two ways: (a) using the Convolution Theorem, (b) using partial fractions. $$F(s)=\frac{1}{s+2}, \quad G(s)=\frac{s+2}{s^{2}+4 s+13}$$

Determine the Laplace transform of \(f\). $$f(t)=e^{3 t} \cos 4 t$$.

Determine the inverse Laplace transform of \(F.\) $$F(s)=\frac{e^{-s}}{(s+1)(s-4)}$$.

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime \prime}-y=6 \cos t, \quad y(0)=0, \quad y^{\prime}(0)=4\).

Use the linearity of \(L\) and the formulas derived in this section to determine \(L[f]\). $$f(t)=3 t^{2}-5 \cos 2 t+\sin 3 t$$

Determine the inverse Laplace transform of the given function. $$F(s)=\frac{1}{s(s+1)}.$$

Use properties of the Laplace transform and the table of Laplace transforms to determine \(L[f]\). $$f(t)=\int_{0}^{t}(t-w)^{2} e^{w} d w$$

Determine \(L^{-1}[F(s) G(s)]\) in the following two ways: (a) using the Convolution Theorem, (b) using partial fractions. $$F(s)=\frac{1}{s^{2}+9}, \quad G(s)=\frac{2}{s^{3}}$$

Determine the Laplace transform of \(f\). $$f(t)=e^{-4 t} \sin 5 t$$.

Determine the inverse Laplace transform of \(F.\) $$F(s)=\frac{e^{-2 s}}{s^{2}+2 s+2}$$.

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime \prime}-9 y=13 \sin 2 t, \quad y(0)=3, \quad y^{\prime}(0)=1\).

Use the linearity of \(L\) and the formulas derived in this section to determine \(L[f]\). $$f(t)=3 t^{2}-5 \cos 2 t+\sin 3 t$$

Determine the inverse Laplace transform of the given function. $$F(s)=\frac{s+4}{(s-1)(s+2)(s-3)}.$$

Determine a function \(f(t)\) that has the given Laplace transform \(F(s)\). $$F(s)=\frac{3}{s^{2}}$$

Determine \(L^{-1}[F(s) G(s)]\) in the following two ways: (a) using the Convolution Theorem, (b) using partial fractions. $$F(s)=\frac{1}{s^{2}}, \quad G(s)=\frac{e^{-\pi s}}{s^{2}+1}$$

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime \prime}-y=8 \sin t-6 \cos t, \quad y(0)=2, \quad y^{\prime}(0)=-1\).

Use the linearity of \(L\) and the formulas derived in this section to determine \(L[f]\). $$f(t)=2 e^{-3 t}+4 e^{t}-5 \sin t$$

Determine the inverse Laplace transform of the given function. $$F(s)=\frac{2 s+3}{(s-2)\left(s^{2}+1\right)}.$$

Determine the Laplace transform of \(f\). $$f(t)=t e^{2 t}$$.

Determine the inverse Laplace transform of \(F.\) $$F(s)=\frac{e^{-s}(s+6)}{s^{2}+9}$$.

Determine a function \(f(t)\) that has the given Laplace transform \(F(s)\). $$F(s)=\frac{4 s+5}{s^{2}+9}$$

Express \(L^{-1}[F(s) G(s)]\) in terms of a convolution integral. $$F(s)=\frac{4}{s^{3}}, \quad G(s)=\frac{s-1}{s^{2}-2 s+5}$$

Determine the inverse Laplace transform of \(F.\) $$F(s)=\frac{e^{-5 s}}{s^{2}+16}$$.

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime \prime}-y^{\prime}-2 y=10 \cos t, \quad y(0)=0, \quad y^{\prime}(0)=-1\).

Determine the inverse Laplace transform of the given function. $$F(s)=\frac{2 s+3}{\left(s^{2}+4\right)\left(s^{2}+1\right)}.$$