Fundamentals of Differential Equations and Boundary Value Problems
7th Edition
ISBN: 9780321977106
Author: Nagle, R. Kent
Publisher: Pearson Education, Limited
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Chapter 7.5, Problem 41E
To determine
The error for the automatic pilot with mild damping.
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Question 1. (10 points)
A researcher is studying tumours in mice. The growth rate for the volume of the tumour V(t) in cm³ is given by
dV
=
1.45V(2 In(V+1)).
dt
(a) (4 pts) Find all the equilibria and determine their stability using the stability condition.
(b) (2 pts) Draw the phase plot f(V) versus V where f(V) = V'. You may find it helpful to use Desmos or Wolfram Alpha to plot the graph of
f(V) versus V (both are free to use online), or you can plot it by hand if you like. On the plot identify each equilibrium as stable or unstable.
(c) (4 pts) Draw direction arrows for the case where the tumour starts at size 3cm³ and for the case where the tumour starts at size 9cm³. Explain
in biological terms what happens to the size of each of these tumours at time progresses.
For the system consisting of the two planes:plane 1: -x + y + z = 0plane 2: 3x + y + 3z = 0a) Are the planes parallel and/or coincident? Justify your answer. What does this tell you about the solution to the system?b) Solve the system (if possible). Show a complete solution. If there is a line of intersection express it in parametric form.
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Chapter 7 Solutions
Fundamentals of Differential Equations and Boundary Value Problems
Ch. 7.2 - In Problems 1-12, use Definition 1 to determine...Ch. 7.2 - In Problems 1-12, use Definition 1 to determine...Ch. 7.2 - In Problems 1-12, use Definition 1 to determine...Ch. 7.2 - In Problems 1-12, use Definition 1 to determine...Ch. 7.2 - In Problems 1-12, use Definition 1 to determine...Ch. 7.2 - In Problems 1-12, use Definition 1 to determine...Ch. 7.2 - In Problems 1-12, use Definition 1 to determine...Ch. 7.2 - In Problems 1-12, use Definition 1 to determine...Ch. 7.2 - In Problems 1-12, use Definition 1 to determine...Ch. 7.2 - In Problems 1 -12, Use Definition 1 to determine...
Ch. 7.2 - In Problems 112, use Definition 1 to determine the...Ch. 7.2 - In Problems 112, use Definition 1 to determine the...Ch. 7.2 - In Problems 13-20, use the Laplace transform table...Ch. 7.2 - Prob. 14ECh. 7.2 - In Problems 13-20, use the Laplace transform table...Ch. 7.2 - In Problems 13-20, use the Laplace transform table...Ch. 7.2 - In Problems 13-20, use the Laplace transform table...Ch. 7.2 - In Problems 13-20, use the Laplace transform table...Ch. 7.2 - In Problems 13-20, use the Laplace transform table...Ch. 7.2 - In Problems 13-20, use the Laplace transform table...Ch. 7.2 - In Problems 2128, determine whether f(t) is...Ch. 7.2 - Prob. 22ECh. 7.2 - In Problems 21-28, determine whether f(t) is...Ch. 7.2 - In Problems 21-28, determine whether f(t) is...Ch. 7.2 - In Problems 21-28, determine whether f(t) is...Ch. 7.2 - In Problems 21-28, determine whether f(t) is...Ch. 7.2 - In Problems 21-28, determine whether f(t) is...Ch. 7.2 - In Problems 21-28, determine whether f(t) is...Ch. 7.2 - Which of the following functions are of...Ch. 7.2 - For the transforms F(s) in Table 7.1, what can be...Ch. 7.2 - Thanks to Eulers formula page 166 and the...Ch. 7.2 - Prob. 32ECh. 7.2 - Prove that if f is piecewise continuous on [a,b]...Ch. 7.3 - In Problems 1- 20, determine the Laplace transform...Ch. 7.3 - In Problems 1-20, determine the Laplace transform...Ch. 7.3 - In Problems 1-20, determine the Laplace transform...Ch. 7.3 - In Problems 1-20, determine the Laplace transform...Ch. 7.3 - In Problems 1-20, determine the Laplace transform...Ch. 7.3 - Prob. 6ECh. 7.3 - In Problems 1-20, determine the Laplace transform...Ch. 7.3 - In Problems 1-20, determine the Laplace transform...Ch. 7.3 - Prob. 9ECh. 7.3 - In Problems 1-20, determine the Laplace transform...Ch. 7.3 - In Problems 1- 20, determine the Laplace transform...Ch. 7.3 - In Problems 1- 20, determine the Laplace transform...Ch. 7.3 - In Problems 1- 20, determine the Laplace transform...Ch. 7.3 - In Problems 1- 20, determine the Laplace transform...Ch. 7.3 - In Problems 1- 20, determine the Laplace transform...Ch. 7.3 - In Problems 1- 20, determine the Laplace transform...Ch. 7.3 - In Problems 1- 20, determine the Laplace transform...Ch. 7.3 - In Problems 1- 20, determine the Laplace transform...Ch. 7.3 - Prob. 19ECh. 7.3 - Prob. 20ECh. 7.3 - Given that L{cosbt}(s)=s/(s2+b2), use the...Ch. 7.3 - Starting with the transform L{1}(s)=1/s, use...Ch. 7.3 - Use Theorem 4 on page 362 to show how entry 32...Ch. 7.3 - Show that L{eattn}(s)=n!/(sa)n+1 in two ways: a....Ch. 7.3 - Use formula (6) to help determine. a. L{tcosbt}....Ch. 7.3 - Prob. 26ECh. 7.3 - Prob. 27ECh. 7.3 - Prob. 28ECh. 7.3 - The transfer function of a linear system is...Ch. 7.3 - Prob. 30ECh. 7.3 - Prob. 31ECh. 7.3 - Prob. 32ECh. 7.3 - Prob. 33ECh. 7.3 - Prob. 34ECh. 7.3 - Prob. 35ECh. 7.3 - Prob. 36ECh. 7.3 - Initial value theorem. Apply the relation...Ch. 7.3 - Prob. 38ECh. 7.4 - In Problems 1-10, determine the inverse Laplace...Ch. 7.4 - In Problems 1-10, determine the inverse Laplace...Ch. 7.4 - In Problems 1-10, determine the inverse Laplace...Ch. 7.4 - In Problems 1-10, determine the inverse Laplace...Ch. 7.4 - Prob. 5ECh. 7.4 - In Problems 1-10, determine the inverse Laplace...Ch. 7.4 - In Problems 1-10, determine the inverse Laplace...Ch. 7.4 - In Problems 1-10, determine the inverse Laplace...Ch. 7.4 - In Problems 1-10, determine the inverse Laplace...Ch. 7.4 - In Problems 1-10, determine the inverse Laplace...Ch. 7.4 - Prob. 11ECh. 7.4 - Prob. 12ECh. 7.4 - In Problems 11-20, determine the partial fraction...Ch. 7.4 - Prob. 14ECh. 7.4 - In Problems 11-20, determine the partial fraction...Ch. 7.4 - In Problems 11-20, determine the partial fraction...Ch. 7.4 - Prob. 17ECh. 7.4 - Prob. 18ECh. 7.4 - In Problems 11-20, determine the partial fraction...Ch. 7.4 - In Problems 11-20, determine the partial fraction...Ch. 7.4 - In Problems 21-30, determine L1{F}....Ch. 7.4 - Prob. 22ECh. 7.4 - In Problems 21-30, determine L1{F}....Ch. 7.4 - In Problems 21-30, determine L1{F}....Ch. 7.4 - In Problems 21-30, determine L1{F}....Ch. 7.4 - In Problems 21-30, determine L1{F}....Ch. 7.4 - In Problems 21-30, determine L1{F}....Ch. 7.4 - In Problems 21-30, determine L1{F}....Ch. 7.4 - In Problems 21-30, determine L1{F}....Ch. 7.4 - In Problems 21-30, determine L1{F}....Ch. 7.4 - Determine the Laplace transform of each of the...Ch. 7.4 - Prob. 32ECh. 7.4 - Theorem 6 in Section 7.3 on page 364 can be...Ch. 7.4 - Theorem 6 in Section 7.3 on page 364 can be...Ch. 7.4 - Theorem 6 in Section 7.3 on page 364 can be...Ch. 7.4 - Theorem 6 in Section 7.3 on page 364 can be...Ch. 7.4 - Prob. 37ECh. 7.4 - Prob. 38ECh. 7.4 - Prob. 39ECh. 7.4 - Heavisides Expansion Formula. Let P(s) and Q(s) be...Ch. 7.4 - Prob. 41ECh. 7.4 - Prob. 42ECh. 7.4 - Prob. 43ECh. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 114, solve the given initial value...Ch. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 114, solve the given initial value...Ch. 7.5 - In Problems 1524, solve for Y(s), the Laplace...Ch. 7.5 - In Problems1524, solve for Y(s), the Laplace...Ch. 7.5 - In Problems 1524, solve for Y(s), the Laplace...Ch. 7.5 - In Problems 1524, solve for Y(s), the Laplace...Ch. 7.5 - In Problems 1524, solve for Y(s), the Laplace...Ch. 7.5 - In Problems 1524, solve for Y(s), the Laplace...Ch. 7.5 - In Problems 1524, solve for Y(s), the Laplace...Ch. 7.5 - In Problems 1524, solve for Y(s), the Laplace...Ch. 7.5 - In Problems 1524, solve for Y(s), the Laplace...Ch. 7.5 - In Problems 1524, solve for Y(s), the Laplace...Ch. 7.5 - In Problems 2528, solve the given third-order...Ch. 7.5 - In Problems 2528, solve the given third-order...Ch. 7.5 - In Problems 2528, solve the given third-order...Ch. 7.5 - In Problems 2528, solve the given third-order...Ch. 7.5 - In Problems 2932, use the method of Laplace...Ch. 7.5 - In Problems 2932, use the method of Laplace...Ch. 7.5 - In Problems 29-32, use the method of Laplace...Ch. 7.5 - In Problems 29-32, use the method of Laplace...Ch. 7.5 - Prob. 33ECh. 7.5 - Use Theorem 6 in Section 7.3, page 364, to show...Ch. 7.5 - In Problems 3538, find solutions to given initial...Ch. 7.5 - In Problems 3538, find solutions to given initial...Ch. 7.5 - In Problems 3538, find solutions to given initial...Ch. 7.5 - In Problems 3538, find solutions to given initial...Ch. 7.5 - Determine the error e(t) for the automatic pilot...Ch. 7.5 - Prob. 40ECh. 7.5 - Prob. 41ECh. 7.6 - In Problems 14, sketch the graph of the given...Ch. 7.6 - Prob. 2ECh. 7.6 - In Problems 14, sketch the graph of the given...Ch. 7.6 - Prob. 4ECh. 7.6 - In Problems 510, express the given function using...Ch. 7.6 - In Problems 510, express the given function using...Ch. 7.6 - Prob. 7ECh. 7.6 - In Problems 5-10, express the given function using...Ch. 7.6 - Prob. 9ECh. 7.6 - Prob. 10ECh. 7.6 - In Problems 1118, determine an inverse Laplace...Ch. 7.6 - Prob. 12ECh. 7.6 - In Problems 1118, determine an inverse Laplace...Ch. 7.6 - In Problems 1118, determine an inverse Laplace...Ch. 7.6 - In Problems 1118, determine an inverse Laplace...Ch. 7.6 - Prob. 16ECh. 7.6 - In Problems 1118, determine an inverse Laplace...Ch. 7.6 - In Problems 1118, determine an inverse Laplace...Ch. 7.6 - The current I(t) in an RLC series circuit is...Ch. 7.6 - The current I(t) in an LC series circuit is...Ch. 7.6 - In Problems 2124, solve the given initial value...Ch. 7.6 - In Problems 2124, solve the given initial value...Ch. 7.6 - In Problems 2124, solve the given initial value...Ch. 7.6 - In Problems 2124, solve the given initial value...Ch. 7.6 - In Problems 25-32, solve the given initial value...Ch. 7.6 - In Problems 2532, solve the given initial value...Ch. 7.6 - In Problems 2532, solve the given initial value...Ch. 7.6 - In Problems 2532, solve the given initial value...Ch. 7.6 - In Problems 2532, solve the given initial value...Ch. 7.6 - In Problems 25-32, solve the given initial value...Ch. 7.6 - In Problems 2532, solve the given initial value...Ch. 7.6 - In Problems 25-32, solve the given initial value...Ch. 7.6 - Prob. 35ECh. 7.7 - In Problems 1-4, determine L{f}, where f(t) is...Ch. 7.7 - Prob. 2ECh. 7.7 - Prob. 3ECh. 7.7 - In Problems 1-4, determine L{f}, where f(t) is...Ch. 7.7 - Prob. 5ECh. 7.7 - Prob. 6ECh. 7.7 - In Problems 5-8, determine L{f}, where the...Ch. 7.7 - Prob. 8ECh. 7.7 - Show that if L{g}(s)=[(s+)(1eTs)]1, where T0 is...Ch. 7.7 - Prob. 10ECh. 7.7 - Prob. 11ECh. 7.7 - Prob. 12ECh. 7.7 - Prob. 14ECh. 7.7 - Prob. 15ECh. 7.7 - Prob. 16ECh. 7.7 - In Problems 1518, find a Taylor series for f(t)...Ch. 7.7 - Prob. 18ECh. 7.7 - Prob. 19ECh. 7.7 - Use the recursive relation (7) and the fact that...Ch. 7.7 - Prob. 21ECh. 7.7 - Prob. 22ECh. 7.7 - Prob. 23ECh. 7.7 - Use the procedure discussed inProblem 23 to show...Ch. 7.7 - Find an expansion for ln[1+(1/s2)] in powers of...Ch. 7.7 - Prob. 26ECh. 7.7 - Prob. 27ECh. 7.8 - In Problems 14, use the convolution theorem to...Ch. 7.8 - Prob. 2ECh. 7.8 - Prob. 3ECh. 7.8 - Prob. 4ECh. 7.8 - Prob. 5ECh. 7.8 - Prob. 6ECh. 7.8 - Prob. 7ECh. 7.8 - In Problems 512, use the convolution theorem to...Ch. 7.8 - Prob. 9ECh. 7.8 - In Problems 512, use the convolution theorem to...Ch. 7.8 - In Problems 512, use the convolution theorem to...Ch. 7.8 - In Problems 512, use the convolution theorem to...Ch. 7.8 - Prob. 13ECh. 7.8 - Find the Laplace transform of f(t):=0tevsin(tv)dvCh. 7.8 - Prob. 15ECh. 7.8 - In Problems 1522, solve the given integral...Ch. 7.8 - Prob. 17ECh. 7.8 - Prob. 18ECh. 7.8 - In Problems 1522, solve the given integral...Ch. 7.8 - In Problems 1522, solve the given integral...Ch. 7.8 - In Problems 1522, solve the given integral...Ch. 7.8 - Prob. 22ECh. 7.8 - In Problems 2328, a linear system is governed by...Ch. 7.8 - Prob. 24ECh. 7.8 - In Problems 2328, a linear system is governed by...Ch. 7.8 - In Problems 2328, a linear system is governed by...Ch. 7.8 - In Problems 2328, a linear system is governed by...Ch. 7.8 - In Problems 2328, a linear system is governed by...Ch. 7.8 - Prob. 29ECh. 7.8 - In Problems 29 and 30, the current I(t) in an RLC...Ch. 7.8 - Prob. 31ECh. 7.8 - Prob. 32ECh. 7.8 - Prob. 33ECh. 7.8 - Prob. 34ECh. 7.8 - Prob. 35ECh. 7.8 - Prob. 36ECh. 7.9 - In Problems 1-6, evaluate the given integral....Ch. 7.9 - Prob. 2ECh. 7.9 - In Problems 1-6, evaluate the given integral....Ch. 7.9 - Prob. 4ECh. 7.9 - In Problems 1-6, evaluate the given integral....Ch. 7.9 - In Problems 1-6, evaluate the given integral....Ch. 7.9 - In Problems 7-12, determine the Laplace transform...Ch. 7.9 - In Problems 7-12, determine the Laplace transform...Ch. 7.9 - In Problems 7-12, determine the Laplace transform...Ch. 7.9 - In Problems 7-12, determine the Laplace transform...Ch. 7.9 - Prob. 11ECh. 7.9 - In Problems 7-12, determine the Laplace transform...Ch. 7.9 - Prob. 13ECh. 7.9 - In Problems 13-20, solve the given symbolic...Ch. 7.9 - Prob. 15ECh. 7.9 - In Problems 13-20, solve the given symbolic...Ch. 7.9 - In Problems 13-20, solve the given symbolic...Ch. 7.9 - In Problems 13-20, solve the given symbolic...Ch. 7.9 - Prob. 19ECh. 7.9 - In Problems 13-20, solve the given symbolic...Ch. 7.9 - In Problems 21-24, solve the given symbolic...Ch. 7.9 - Prob. 22ECh. 7.9 - In Problems 21-24, solve the given symbolic...Ch. 7.9 - Prob. 24ECh. 7.9 - Prob. 25ECh. 7.9 - Prob. 26ECh. 7.9 - Prob. 27ECh. 7.9 - Prob. 28ECh. 7.9 - Prob. 29ECh. 7.9 - Prob. 30ECh. 7.9 - A linear system is said to be stable if its...Ch. 7.9 - A linear system is said to be asymptotically...Ch. 7.9 - Prob. 33ECh. 7.9 - Prob. 34ECh. 7.9 - Figure 7.29 shows a beam of length 2 that is...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - Prob. 9ECh. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - Prob. 17ECh. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - Use the method of Laplace transforms to solve...Ch. 7.10 - Recompute the coupled mass-spring oscillator...Ch. 7.10 - In Problems 23 and 24, find a system of...Ch. 7.10 - In Problems 23 and 24, find a system of...Ch. 7.RP - In Problems 1 and 2, use the definition of the...Ch. 7.RP - In Problems 1 and 2, use the definition of the...Ch. 7.RP - Prob. 3RPCh. 7.RP - In Problems 3-10, determine the Laplace transform...Ch. 7.RP - In Problems 3-10, determine the Laplace transform...Ch. 7.RP - In Problems 3-10, determine the Laplace transform...Ch. 7.RP - Prob. 7RPCh. 7.RP - Prob. 8RPCh. 7.RP - Prob. 9RPCh. 7.RP - Prob. 10RPCh. 7.RP - Prob. 11RPCh. 7.RP - In Problems 11-17, determine the inverse Laplace...Ch. 7.RP - Prob. 13RPCh. 7.RP - Prob. 14RPCh. 7.RP - Prob. 15RPCh. 7.RP - Prob. 16RPCh. 7.RP - Prob. 17RPCh. 7.RP - Prob. 18RPCh. 7.RP - Prob. 19RPCh. 7.RP - In Problems 19-24, solve the given initial value...Ch. 7.RP - Prob. 21RPCh. 7.RP - Prob. 22RPCh. 7.RP - Prob. 23RPCh. 7.RP - Prob. 24RPCh. 7.RP - In Problems 25 and 26, find solutions to the given...Ch. 7.RP - In Problems 25 and 26, find solutions to the given...Ch. 7.RP - Prob. 27RPCh. 7.RP - Prob. 28RPCh. 7.RP - A linear system is governed by y5y+6y=g(t). Find...Ch. 7.RP - Prob. 30RPCh. 7.RP - Prob. 31RPCh. 7.RP - In Problems 31 and 32, use Laplace transforms to...
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