Figure 15.26

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• 15.7 Find the Laplace transform of the following signals:

$$(a) f(t) = (2t + 4)u(t)$$

(b)

$$g(t) = (4 + 3e^{-2t})u(t)$$

• (c) h(t) = (6 sin(3t) + 8 cos(3t))u(t)
• (d) x(t) = (e⁻2*^t* cosh(4t))u(t)
• 15.8 Find the Laplace transform F(s), given that f(t) is:
  • (a) 2tu(t − 4) (b) 5 cos(t) δ(t − 2) (c) e^−t u(t − t) (d) sin(2t)u(t − τ)
• 15.9 Determine the Laplace transforms of these functions:

(a)

$$f(t) = (t - 4)u(t - 2)$$

\n(b) $g(t) = 2e^{-4t}u(t - 1)$
\n(c) $h(t) = 5 \cos(2t - 1)u(t)$

• (d) p(t) = 6[u(t − 2) − u(t − 4)]
• 15.10 In two different ways, find the Laplace transform of

$$g(t) = \frac{d}{dt}(te^{-t}\cos t)$$

15.11 Find F(s) if:

(a)

$$f(t) = 6e^{-t} \cosh 2t$$

(b) $f(t) = 3te^{-2t} \sinh 4t$
(c) $f(t) = 8e^{-3t} \cosh tu(t - 2)$

• 15.12 If g(t) = 4e⁻2*^t* cos 4t, find G(s).
• 15.13 Find the Laplace transform of the following functions: (a) t cos tu(t) (b) e^−t t sin tu(t)

(a)

$$t \cos u(t)$$

(c) $\frac{\sin \beta t}{t} u(t)$

t

15.14 Find the Laplace transform of the signal in Fig. 15.26.

Figure 15.26

For Prob. 15.14.

15.15 Determine the Laplace transform of the function in Fig. 15.27.

For Prob. 15.15.

15.16 Obtain the Laplace transform of f(t) in Fig. 15.28.

Figure 15.29

For Prob. 15.17.

15.18 Obtain the Laplace transforms of the functions in Fig. 15.30.

Figure 15.30

For Prob. 15.18.

15.19 Calculate the Laplace transform of the infinite train of unit impulses in Fig. 15.31.

Figure 15.31 For Prob. 15.19.

15.17 Using Fig. 15.29, design a problem to help other students better understand the Laplace transform of a simple, non-periodic waveshape.

Figure 15.32 For Prob. 15.20.

15.21 Obtain the Laplace transform of the periodic waveform in Fig. 15.33.

Figure 15.33

For Prob. 15.21.

15.22 Find the Laplace transforms of the functions in Fig. 15.34.

15.23 Determine the Laplace transforms of the periodic functions in Fig. 15.35.

Figure 15.35

For Prob. 15.23.

15.24 Design a problem to help other students better understand how to find the initial and final value of a transfer function.

15.25 Let

$$F(s) = \frac{18(s+1)}{(s+2)(s+3)}$$

• (a) Use the initial and final value theorems to find f(0) and f (∞).
• (b) Verify your answer in part (a) by finding f(t), using partial fractions.

15.26 Determine the initial and final values of f(t), if they exist, given that:

(a)

$$F(s) = \frac{5s^2 + 3}{s^3 + 4s^2 + 6}$$

\n(b) $F(s) = \frac{s^2 - 2s + 1}{4(s - 2)(s^2 + 2s + 4)}$

Section 15.4 The Inverse Laplace Transform

15.27 Determine the inverse Laplace transform of each of the following functions:

Problems 709
\nDetermine the initial and final values of

$$f(t)$$

, if they exist, given that:
\n(a) $F(s) = \frac{s^2 + 3}{s^3 + 4s^2 + 6}$
\n(b) $F(s) = \frac{s^2 - 2s + 1}{4(s - 2)(s^2 + 2s + 4)}$
\nOn 15.4 The Inverse Laplace Transform
\n7 Determine the inverse Laplace transform of each of the following functions:
\n(a) $F(s) = \frac{1}{s} + \frac{2}{s + 1}$
\n(b) $G(s) = \frac{3s + 1}{s + 4}$
\n(c) $H(s) = \frac{12}{(s + 1)(s + 3)}$
\n(d) $J(s) = \frac{12}{(s + 2)^2(s + 4)}$
\nDesign a problem to help other students better understand how to find the inverse Laplace transform.
\nFind the inverse Laplace transform of:
\n $F(s) = \frac{s^2 + 2}{s^3 + 2s^2 + 2s}$
\nFind the inverse Laplace transform of:
\n(a) $F_1(s) = \frac{6s^2 + 8s + 3}{s(s^2 + 2s + 5)}$
\n(b) $F_2(s) = \frac{s^2 + 5s + 6}{(s + 1)^2(s + 4)}$
\n(c) $F_3(s) = \frac{10}{(s + 1)(s^2 + 4s + 8)}$
\nFind $f(t)$ for each $F(s)$ :
\n(a) $\frac{10s}{(s + 1)(s + 2)^3}$
\n(b) $\frac{2s^2 + 4s + 1}{(s + 1)(s + 2)^3}$
\n(c) $\frac{s + 1}{(s + 2)(s^2 + 2s + 5)}$
\nDetermine the inverse Laplace transform of each of the following functions:
\n(a) $\frac{8(s + 1)(s + 2)}{s(s + 2)(s + 4)}$
\n(b) $\frac{s^2 - 2s + 4}{s(s + 1)(s + 2)^2}$
\n(c) $\frac{s^2 + 1}{(s + 1)(s + 2)^2}$

• 15.28 Design a problem to help other students better understand how to find the inverse Laplace transform.
• 15.29 Find the inverse Laplace transform of:

$$F(s) = \frac{s^2 + 2}{s^3 + 2s^2 + 2s}$$

15.30 Find the inverse Laplace transform of:

Find the inverse Laplace transform
\n(a)

$$F_1(s) = \frac{6s^2 + 8s + 3}{s(s^2 + 2s + 5)}$$

\n(b) $F_2(s) = \frac{s^2 + 5s + 6}{(s+1)^2(s+4)}$
\n(c) $F_3(s) = \frac{10}{(s+1)(s^2 + 4s + 8)}$

15.31 Find f(t) for each F(s):

Find

$$f(t)$$

for each $F(s)$ :
\n(a) $\frac{10s}{(s + 1)(s + 2)(s + 3)}$
\n(b) $\frac{2s^2 + 4s + 1}{(s + 1)(s + 2)^3}$
\n(c) $\frac{s + 1}{(s + 2)(s^2 + 2s + 5)}$

15.32 Determine the inverse Laplace transform of each of the following functions:

the following functions:
\n(a)

$$\frac{8(s + 1)(s + 3)}{s(s + 2)(s + 4)}$$

\n(b)

$$\frac{s^2 - 2s + 4}{(s + 1)(s + 2)^2}$$

\n(c)

$$\frac{s^2 + 1}{(s + 3)(s^2 + 4s + 5)}$$

15.33 Calculate the inverse Laplace transform of:

(a)

$$\frac{6(s-1)}{s^4 - 1}$$

(b) $\frac{se^{-\pi s}}{s^2 + 1}$
(c) $\frac{8}{s(s+1)^3}$

15.34 Find the time functions that have the following Laplace transforms:

(a)

$$F(s) = 10 + \frac{s^2 + 1}{s^2 + 4}$$

\n(b) $G(s) = \frac{e^{-s} + 4e^{-2s}}{s^2 + 6s + 8}$
\n(c) $H(s) = \frac{(s + 1)e^{-2s}}{s(s + 3)(s + 4)}$

15.35 Obtain f(t) for the following transforms:

Obtain

$$f(t)$$

for the following
\n(a) $F(s) = \frac{(s+3)e^{-6s}}{(s+1)(s+2)}$
\n(b) $F(s) = \frac{4 - e^{-2s}}{s^2 + 5s + 4}$
\n(c) $F(s) = \frac{se^{-s}}{(s+3)(s^2 + 4)}$

15.36 Obtain the inverse Laplace transforms of the following functions:

following functions:
\n(a)

$$X(s) = \frac{3}{s^2(s+2)(s+3)}$$

\n(b) $Y(s) = \frac{2}{s(s+1)^2}$
\n(c) $Z(s) = \frac{5}{s(s+1)(s^2+6s+10)}$

Laplace transform of:
\n15.37 Find the inverse Laplace transform of:
\n**(b)**

$$\frac{se^{-xs}}{s^2 + 1}$$

\n**(c)** $F(s) = \frac{s^2 + 4s + 5}{(s + 2)(s^2 + 2s + 2)}$
\n**(d)** $D(s) = \frac{10s}{(s^2 + 1)(s^2 + 4)}$
\n15.38 Find $f(t)$ given that:
\n**(a)** $F(s) = \frac{10s}{(s^2 + 1)(s^2 + 4)}$
\n15.38 Find $f(t)$ given that:
\n**(a)** $F(s) = \frac{s^2 + 4s}{s^2 + 10s + 26}$
\n**(b)** $F(s) = \frac{5s^2 + 7s + 29}{s(s^2 + 4s + 29)}$
\nIlowing transforms:
\n**(a)** $F(s) = \frac{5s^2 + 7s + 29}{(s^2 + 2s + 17)(s^2 + 4s + 20)}$
\n**(b)** $F(s) = \frac{2s^3 + 4s^2 + 1}{(s^2 + 2s + 17)(s^2 + 6s + 3)}$
\n**(c)** $F(s) = \frac{2s^3 + 4s^2 + 1}{(s^2 + 2s + 17)(s^2 + 6s + 3)}$
\n**(d)** $F(s) = \frac{s^2 + 4}{(s^2 + 2s + 17)(s^2 + 4s + 20)}$
\n**(e)** $F(s) = \frac{s^2 + 4}{(s^2 + 2s + 17)(s^2 + 6s + 3)}$
\n15.40 Show that
\n**(f)** $F(s) = \frac{s^2 + 4}{(s^2 + 2s + 17)(s^2 + 6s + 3)}$
\n**(g)** Show that
\n**(h)** $F(s) = \frac{s^2 + 4}{(s^2 + 2s + 17)(s^2 + 6s + 3)}$
\n**(i)** F(s) = \frac{s^2 + 4}{(s^2 + 2s + 17)(s^2 +

15.38 Find f(t) given that:

Find

$$f(t)$$

given that:
\n(a) $F(s) = \frac{s^2 + 4s}{s^2 + 10s + 26}$
\n(b) $F(s) = \frac{5s^2 + 7s + 29}{s(s^2 + 4s + 29)}$

15.39 Determine f(t) if: *

$$s(s^{2} + 4s + 29)$$

Determine $f(t)$ if:
(a) $F(s) = \frac{2s^{3} + 4s^{2} + 1}{(s^{2} + 2s + 17)(s^{2} + 4s + 20)}$
(b) $F(s) = \frac{s^{2} + 4}{(s^{2} + 9)(s^{2} + 6s + 3)}$

15.40 Show that

at
\n

$$\mathcal{L}^{-1} \left[ \frac{4s^2 + 7s + 13}{(s+2)(s^2 + 2s + 5)} \right] =$$

\n

$$\left[ \sqrt{2}e^{-t} \cos(2t + 45^\circ) + 3e^{-2t} \right] u(t)$$

Section 15.5 The Convolution Integral

15.41 Let x(t) and y(t) be as shown in Fig. 15.36. Find z(t) = x(t) * y(t). *

15.42 Design a problem to help other students better understand how to convolve two functions together.

For Prob. 15.41.

^* An asterisk indicates a challenging problem.

Figure 15.37

• For Prob. 15.43.
• 15.44 Obtain the convolution of the pairs of signals in Fig. 15.38.

15.45 Given h(t) = 4e⁻2*^t u*(t) and x(t) = δ(t) − 2e⁻2*^t u*(t), find y(t) = x(t) * h(t).

15.46 Given the following functions

x(t) = 2δ(t), y(t) = 4u(t), z(t) = e⁻2*^t u*(t), evaluate the following convolution operations.

\n- (a)

$$x(t) \ast y(t)$$

\n- (b) $x(t) \ast z(t)$ \n- (c) $y(t) \ast z(t)$ \n- (d) $y(t) \ast [y(t) + z(t)]$ \n

15.47 A system has the transfer function

as the transfer function

$$H(s) = \frac{6s}{(s+1)(s+2)}$$

• (a) Find the impulse response of the system.
• (b) Determine the output y(t), given that the input is x(t) = u(t).
• 15.48 Find f(t) using convolution given that:

Find

$$f(t)$$

using convolution
\n(a) $F(s) = \frac{4}{(s^2 + 2s + 5)^2}$
\n(b) $F(s) = \frac{2s}{(s + 1)(s^2 + 4)}$

15.49 Use the convolution integral to find: *

(a)

$$t * e^{at}u(t)$$

(b) $\cos(t) * \cos(t)u(t)$

Section 15.6 Application to Integrodifferential Equations

15.50 Use the Laplace transform to solve the differential equation

$$\frac{d^2v(t)}{dt^2} + 2\frac{dv(t)}{dt} + 10v(t) = 3\cos 2t$$

subject to v(0) = 1, dv(0)∕dt = −2.

15.51 Given that v(0) = 5 and dv(0)∕dt = 10, solve

$$\frac{d^2v}{dt^2} + 5\frac{dv}{dt} + 6v = 25e^{-t}u(t)$$

15.52 Use the Laplace transform to find i(t) for t > 0 if

$$\frac{d^2i}{dt^2} + 3\frac{di}{dt} + 2i + \delta(t) = 0,$$ $$i(0) = 0, \qquad i'(0) = 3$$

15.53 Use Laplace transforms to solve for x(t) in *

$$x(t) = \cos t + \int_0^t e^{\lambda - t} x(\lambda) d\lambda$$

15.54 Design a problem to help other students better understand solving second order differential equations with a time varying input.

15.55 Solve for y(t) in the following differential equation if the initial conditions are zero.

$$\frac{d^3y}{dt^3} + 6\frac{d^2y}{dt^2} + 8\frac{dy}{dt} = e^{-t}\cos 2t$$

15.56 Solve for v(t) in the integrodifferential equation

$$12\frac{dv}{dt} + 36 \int_0^t v \, d\tau = 0$$

given that v(0) = 2.

15.57 Design a problem to help other students better understand solving integrodifferential equations with a periodic input, using Laplace transforms.

15.58 Given that

$$\frac{dv}{dt} + 2v + 5 \int_0^t v(\lambda) d\lambda = 4u(t)$$

with v(0) = −1, determine v(t) for t > 0.

15.59 Solve the integrodifferential equation

$$\frac{dy}{dt} + 4y + 3 \int_0^t y \, d\tau = 18e^{-2t} u(t), \qquad y(0) = -3$$

15.60 Solve the following integrodifferential equation

$$2\frac{dx}{dt} + 5x + 3\int_0^t x\,dt + 4 = \sin 4t, \qquad x(0) = 1$$

• 15.61 Solve the following differential equations subject to the specified initial conditions.
  • (a) d² v/dt² + 4v = 12, v(0) = 0, dv(0)/dt = 2 (b) d² i/dt² + 5di/dt + 4i = 8, i(0) = −1, di(0)/dt = 0 (c) d² v/dt² + 2dv/dt + v = 3, v(0) = 5, dv(0)/dt = 1 (d) d² i/dt² + 2di/dt + 5i = 10, i(0) = 4, di(0)/dt = −2

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