Fundamentals of Differential Equations and Boundary Value Problems
7th Edition
ISBN: 9780321977106
Author: Nagle, R. Kent
Publisher: Pearson Education, Limited
expand_more
expand_more
format_list_bulleted
Question
Chapter 8.5, Problem 15E
To determine
The solution of the initial value problem.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
For each dif erential equation in Problems 1–21, find the general solutionby finding the homogeneous solution and a particular solution.
Please DO NOT YOU THE PI method where 1/f(r) * x. Dont do that.
Instead do this, assume for yp = to something, do the 1 and 2 derivative of it and then plug it in the equation to find the answer.
In Problems 1 through 6, express the solution of the given ini-
tial value problem as a sum of two oscillations as in Eq. (8).
Throughout, primes denote derivatives with respect to time t.
In Problems 1–4, graph the solution function x(t) in such a
way that you can identify and label (as in Fig. 3.6.2) its pe-
riod.
4. x" + 25x = 90 cos 41; x (0) = 0, x'(0) = 90
In Problems 1 through 6, express the solution of the given ini-
tial value problem as a sum of two oscillations as in Eq. (8).
Throughout, primes denote derivatives with respect to time t.
In Problems 1–4, graph the solution function x(t) in such a
way that you can identify and label (as in Fig. 3.6.2) its pe-
riod.
3. x" + 100x = 225 cos 5t + 300 sin 5t; x(0) = 375, x'(0) = 0
Chapter 8 Solutions
Fundamentals of Differential Equations and Boundary Value Problems
Ch. 8.1 - In Problems 18, determine the first three nonzero...Ch. 8.1 - In Problems 18, determine the first three nonzero...Ch. 8.1 - In Problems 18, determine the first three nonzero...Ch. 8.1 - In Problems 18, determine the first three nonzero...Ch. 8.1 - In Problems 18, determine the first three nonzero...Ch. 8.1 - In Problems 1-8, determine the first three nonzero...Ch. 8.1 - In Problems 1-8, determine the first three nonzero...Ch. 8.1 - In Problems 1-8, determine the first three nonzero...Ch. 8.1 - a. Construct the Taylor polynomial p3(x) of degree...Ch. 8.1 - a. Construct the Taylor polynomial p3(x) of degree...
Ch. 8.1 - Prob. 11ECh. 8.1 - Prob. 12ECh. 8.1 - Duffings Equation. In the study of a nonlinear...Ch. 8.1 - Soft versus Hard Springs. For Duffings equation...Ch. 8.1 - Prob. 15ECh. 8.1 - van der Pol Equation. In the study of the vacuum...Ch. 8.2 - In Problems 1-6, determine the convergence set of...Ch. 8.2 - In Problems 1-6, determine the convergence set of...Ch. 8.2 - In Problems 1-6, determine the convergence set of...Ch. 8.2 - In Problems 1-6, determine the convergence set of...Ch. 8.2 - In Problems 1-6, determine the convergence set of...Ch. 8.2 - In Problems 1-6, determine the convergence set of...Ch. 8.2 - Prob. 7ECh. 8.2 - Determine the convergence set of the given power...Ch. 8.2 - In Problems 9 and 10, find the power series...Ch. 8.2 - In Problems 9 and 10, find the power series...Ch. 8.2 - In Problems 11-14, find the first three nonzero...Ch. 8.2 - In Problems 11-14, find the first three nonzero...Ch. 8.2 - Prob. 13ECh. 8.2 - In Problems 11-14, find the first three nonzero...Ch. 8.2 - Prob. 15ECh. 8.2 - Prob. 16ECh. 8.2 - Prob. 17ECh. 8.2 - In Problems 17-20, find a power series expansion...Ch. 8.2 - Prob. 19ECh. 8.2 - In Problems 17-20, find a power series expansion...Ch. 8.2 - Prob. 21ECh. 8.2 - In Problems 21 and 22, find a power series...Ch. 8.2 - Prob. 23ECh. 8.2 - In Problems 23-26, express the given power series...Ch. 8.2 - Prob. 25ECh. 8.2 - In Problems 23-26, express the given power series...Ch. 8.2 - Prob. 27ECh. 8.2 - Show that...Ch. 8.2 - In Problems 29-34, determine the Taylor series...Ch. 8.2 - In Problems 2934, determine the Taylor series...Ch. 8.2 - Prob. 31ECh. 8.2 - In Problems 2934, determine the Taylor series...Ch. 8.2 - Prob. 33ECh. 8.2 - In Problems 2934, determine the Taylor series...Ch. 8.2 - Prob. 35ECh. 8.2 - Let f(x) and g(x) be analytic at x0. Determine...Ch. 8.2 - Prob. 37ECh. 8.2 - Prob. 38ECh. 8.3 - In Problems 110, determine all the singular points...Ch. 8.3 - In Problems 110, determine all the singular points...Ch. 8.3 - In Problems 110, determine all the singular points...Ch. 8.3 - Prob. 4ECh. 8.3 - In Problem 110, determine all the singular points...Ch. 8.3 - Prob. 6ECh. 8.3 - Prob. 7ECh. 8.3 - In Problems 110, determine all the singular points...Ch. 8.3 - In Problems 110, determine all the singular points...Ch. 8.3 - In Problems 110, determine all the singular points...Ch. 8.3 - In Problems 1118, find at least the first four...Ch. 8.3 - In Problems 1118, find at least the first four...Ch. 8.3 - Prob. 13ECh. 8.3 - In Problems 1118, find at least the first four...Ch. 8.3 - In Problems 1118, find at least the first four...Ch. 8.3 - Prob. 16ECh. 8.3 - In Problems 1118, find at least the first four...Ch. 8.3 - Prob. 18ECh. 8.3 - In Problems 1924, find a power series expansion...Ch. 8.3 - In Problems 1924, find a power series expansion...Ch. 8.3 - In Problems 1924, find a power series expansion...Ch. 8.3 - In Problems 1924, find a power series expansion...Ch. 8.3 - In Problems 1924, find a power series expansion...Ch. 8.3 - In Problems 19-24, find a power series expansion...Ch. 8.3 - In Problems 25-28, find at least the first four...Ch. 8.3 - Prob. 26ECh. 8.3 - In Problems 25-28, find at least the first four...Ch. 8.3 - In Problems 25-28, find at least the first four...Ch. 8.3 - Prob. 29ECh. 8.3 - Prob. 30ECh. 8.3 - In Problems 29-31, use the first few terms of the...Ch. 8.3 - Prob. 32ECh. 8.3 - Use the ratio test to show that the radius of...Ch. 8.3 - Prob. 34ECh. 8.3 - Prob. 35ECh. 8.3 - Variable Spring Constant. As a spring is heated,...Ch. 8.4 - In Problems 16, find a minimum value for the...Ch. 8.4 - In Problems 16, find a minimum value for the...Ch. 8.4 - In Problems 16, find a minimum value for the...Ch. 8.4 - In Problems 16, find a minimum value for the...Ch. 8.4 - Prob. 5ECh. 8.4 - In Problems 16, find a minimum value for the...Ch. 8.4 - In Problems 712, find at least the first four...Ch. 8.4 - In Problems 712, find at least the first four...Ch. 8.4 - In Problems 712, find at least the first four...Ch. 8.4 - Prob. 10ECh. 8.4 - In Problems 712, find at least the first four...Ch. 8.4 - In Problems 712, find at least the first four...Ch. 8.4 - In Problems 1319, find at least the first four...Ch. 8.4 - In Problems 1319, find at least the first four...Ch. 8.4 - In Problems 1319, find at least the first four...Ch. 8.4 - Prob. 16ECh. 8.4 - In Problems 13-19, find at least the first four...Ch. 8.4 - In Problems 13-19, find at least the first four...Ch. 8.4 - In Problems 13-19, find at least the first four...Ch. 8.4 - To derive the general solution given by equations...Ch. 8.4 - In Problems 21-28, use the procedure illustrated...Ch. 8.4 - Prob. 22ECh. 8.4 - In Problems 21-28, use the procedure illustrated...Ch. 8.4 - Prob. 24ECh. 8.4 - In Problems 21-28, use the procedure illustrated...Ch. 8.4 - In Problems 21-28, use the procedure illustrated...Ch. 8.4 - In Problems 21-28, use the procedure illustrated...Ch. 8.4 - Prob. 28ECh. 8.4 - The equation (1x2)y2xy+n(n+1)y=0, where n is an...Ch. 8.4 - Aging Spring. As a spring ages, its spring...Ch. 8.4 - Aging Spring without Damping. In the mass-spring...Ch. 8.5 - Prob. 1ECh. 8.5 - Prob. 2ECh. 8.5 - Prob. 3ECh. 8.5 - Prob. 4ECh. 8.5 - Prob. 5ECh. 8.5 - Prob. 6ECh. 8.5 - Prob. 7ECh. 8.5 - Prob. 8ECh. 8.5 - Prob. 9ECh. 8.5 - Prob. 10ECh. 8.5 - Prob. 11ECh. 8.5 - Prob. 12ECh. 8.5 - Prob. 13ECh. 8.5 - Prob. 14ECh. 8.5 - Prob. 15ECh. 8.5 - Prob. 16ECh. 8.5 - In Problems 15-17, solve the given initial value...Ch. 8.5 - Prob. 18ECh. 8.5 - Prob. 19ECh. 8.6 - In Problems 1-10, classify each singular point...Ch. 8.6 - Prob. 2ECh. 8.6 - Prob. 3ECh. 8.6 - Prob. 4ECh. 8.6 - Prob. 5ECh. 8.6 - Prob. 6ECh. 8.6 - Prob. 7ECh. 8.6 - Prob. 8ECh. 8.6 - Prob. 9ECh. 8.6 - Prob. 10ECh. 8.6 - Prob. 11ECh. 8.6 - In Problems 11-18, find the indicial equation and...Ch. 8.6 - In Problems 11-18, find the indicial equation and...Ch. 8.6 - In Problems 11-18, find the indicial equation and...Ch. 8.6 - In Problems 11-18, find the indicial equation and...Ch. 8.6 - In Problems 1118, find the indicial equation and...Ch. 8.6 - In Problems 1118, find the indicial equation and...Ch. 8.6 - In Problems 1118, find the indicial equation and...Ch. 8.6 - Prob. 19ECh. 8.6 - Prob. 20ECh. 8.6 - Prob. 21ECh. 8.6 - Prob. 22ECh. 8.6 - Prob. 23ECh. 8.6 - Prob. 24ECh. 8.6 - Prob. 25ECh. 8.6 - Prob. 26ECh. 8.6 - Prob. 27ECh. 8.6 - Prob. 28ECh. 8.6 - Prob. 29ECh. 8.6 - Prob. 30ECh. 8.6 - Prob. 31ECh. 8.6 - Prob. 32ECh. 8.6 - Prob. 33ECh. 8.6 - Prob. 34ECh. 8.6 - Prob. 35ECh. 8.6 - Prob. 36ECh. 8.6 - Prob. 37ECh. 8.6 - Prob. 38ECh. 8.6 - In Problems 39 and 40, try to use the method of...Ch. 8.6 - Prob. 40ECh. 8.6 - Prob. 41ECh. 8.6 - Prob. 42ECh. 8.6 - Prob. 43ECh. 8.6 - Prob. 44ECh. 8.6 - Prob. 45ECh. 8.6 - Prob. 46ECh. 8.6 - Prob. 47ECh. 8.7 - In Problems 1-14, find at least the first three...Ch. 8.7 - Prob. 2ECh. 8.7 - Prob. 3ECh. 8.7 - Prob. 4ECh. 8.7 - Prob. 5ECh. 8.7 - In Problems 1-14, find at least the first three...Ch. 8.7 - Prob. 7ECh. 8.7 - Prob. 8ECh. 8.7 - In Problems 1-14, find at least the first three...Ch. 8.7 - Prob. 10ECh. 8.7 - Prob. 11ECh. 8.7 - Prob. 12ECh. 8.7 - Prob. 13ECh. 8.7 - Prob. 14ECh. 8.7 - In Problems 15 and 16, determine whether the given...Ch. 8.7 - Prob. 16ECh. 8.7 - In Problems 17-20, find at least the first three...Ch. 8.7 - Prob. 18ECh. 8.7 - In Problems 17-20, find at least the first three...Ch. 8.7 - Prob. 20ECh. 8.7 - Prob. 21ECh. 8.7 - In Problem 21 consider a column with a rectangular...Ch. 8.7 - Prob. 23ECh. 8.7 - Prob. 24ECh. 8.7 - Prob. 25ECh. 8.7 - To obtain two linearly independent solutions to...Ch. 8.8 - In Problems 1-4, express a general solution to the...Ch. 8.8 - Prob. 2ECh. 8.8 - In Problems 1-4, express a general solution to the...Ch. 8.8 - In Problems 1-4, express a general solution to the...Ch. 8.8 - Prob. 5ECh. 8.8 - Prob. 6ECh. 8.8 - Prob. 7ECh. 8.8 - Prob. 8ECh. 8.8 - Prob. 9ECh. 8.8 - Prob. 10ECh. 8.8 - Show that the confluent hypergeometric equation...Ch. 8.8 - Prob. 12ECh. 8.8 - Prob. 13ECh. 8.8 - Prob. 14ECh. 8.8 - Prob. 15ECh. 8.8 - Prob. 16ECh. 8.8 - Prob. 17ECh. 8.8 - Prob. 18ECh. 8.8 - In Problems 19 and 20, a Bessel equation is given....Ch. 8.8 - Prob. 21ECh. 8.8 - Prob. 22ECh. 8.8 - Prob. 23ECh. 8.8 - Prob. 24ECh. 8.8 - Show that J1/2(x)=(2/x)1/2sinx and...Ch. 8.8 - The Bessel functions of order v=n+1/2, n any...Ch. 8.8 - Prob. 27ECh. 8.8 - Prob. 28ECh. 8.8 - Prob. 29ECh. 8.8 - Prob. 30ECh. 8.8 - Prob. 31ECh. 8.8 - To prove Rodriguess formula (52) for Legendre...Ch. 8.8 - Prob. 34ECh. 8.8 - Prob. 35ECh. 8.8 - Prob. 36ECh. 8.8 - The Hermite polynomials Hn(x) are polynomial...Ch. 8.8 - Prob. 38ECh. 8.8 - Prob. 39ECh. 8.8 - Reduction to Bessels Equation. The class of...Ch. 8.8 - a. Show that the substitution z(x)=xy(x) renders...Ch. 8.RP - Find the first four nonzero terms in the Taylor...Ch. 8.RP - Prob. 2RPCh. 8.RP - Find at least the first four nonzero terms in a...Ch. 8.RP - Prob. 4RPCh. 8.RP - Find at least the first four nonzero terms in a...Ch. 8.RP - Prob. 6RPCh. 8.RP - Use the method of Frobenius to find at least the...Ch. 8.RP - Find the indicial equation and its roots and state...Ch. 8.RP - Find at least the first three nonzero terms in the...
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.Similar questions
- In Problems 1 through 6, express the solution of the given ini- tial value problem as a sum of two oscillations as in Eq. (8). Throughout, primes denote derivatives with respect to time t. In Problems 1–4, graph the solution function x(t) in such a way that you can identify and label (as in Fig. 3.6.2) its pe- riod. 1. x" + 9x = 10 cos 2t; x(0) = x'(0) = 0arrow_forwardIn Problems 27 – 36 solve the given initial-value problemarrow_forwardII. Solve the following initial-value problems (IVPS): = = 4(x² + 1), x (7) = 1 (a) dx dtarrow_forward
- In Problems 1 through 6, express the solution of the given ini- tial value problem as a sum of two oscillations as in Eq. (8). Throughout, primes denote derivatives with respect to time t. In Problems 1–4, graph the solution function x(t) in such a way that you can identify and label (as in Fig. 3.6.2) its pe- riod. 5. mx" +kx = Fo cos wt with w # wo; x(0) = xo, x'(0) = 0arrow_forwardProblem 2 a. If y(t) = x₁(1) * x₂ (t)= e(+3)u(t+3), find z(t) = x₁(1+4)*x₂(t-5). b. If x₁ (1) = 8e²¹u(-1) and x₂(t)=0.258(1-4), find y(t)=x (t)* x₂(t). c. If x₁(t) = 4e-2¹u(t) and x₂ (1)= u(1), find y(t) = x₁(1) * x₂ (1)arrow_forward6. S(t + 1)e2 dtarrow_forward
- 5. Consider the equations: det f(x) = A(1+)™* g(x) = A· ex and %D %3D For x =1 (a one year investment), and A = 1 (a $1 investment) we have: %3D n.1 f(1) = 1(1+)" g(1) = 1· e"1 and Simplified: f(1) = (1 + #)" and g(1) = e™arrow_forward2. Two separate rabbit populations are observed for 80 weeks, starting at the same time and with the same initial populations. The growth rates of two rabbit populations are modeled as follows, where t = 0 corresponds to the beginning of the observation period: r₁(t) = 4 sin(1) + 0.11 + 1, where r, is rabbits per week, and t is time in weeks, r₂(t) = 1¹2, where 2 is rabbits per week, and t is time in weeks. Below is a graph of the curves representing the rates of growth of the two populations over the observation period: 11 10 10 20 30 40 50 60 70 80 A. Using your calculator, find (approximately) the first positive time for which the rates of growth for the two populations are the same. B. What's the physical significance of the area between the two curves from time t => 0 until the first time where the two rates are the same? What does the area represent? C. Suppose you want to find the first time (call it 7) after the beginning of the observation period at which the two rabbit…arrow_forwardProblem 4.2 (Textbook: Figure 2.5 on P84): Hypothetically, a patient is diagnosed to have P, coronavirus at time t = 0 and these viruses grow subsequently (nobody really knows the truth but we assume) according to = -aP(M – P) where P(t) is the number of the cells at time t while a and M are two given positive constants. The patient will die when the number of bugs approaches infinity. Consider the case Po > M, find the time the patient has left to live, timed from t = 0. Your result is a formula with the given parameters Pg, M, a. For a more concrete example, please set P, = 1984, M = 1000, a = 1. Compute the patient's survival time. Never mind if the number is small when you use , a = 1 as no unit is given. If you insist having a "reasonable" number, I set a = 10* for which the time unit is week. Either number you use is correct for your HW solution.arrow_forward
- 2. Two separate rabbit populations are observed for 80 weeks, starting at the same time and with the same initial populations. The growth rates of two rabbit populations are modeled as follows, where t = 0 corresponds to the beginning of the observation period: ri(1) = 4 sin(1) + 0. 1t + 1, where r¡ is rabbits per week, and t is time in weeks, r2(t) = t\/2, where r2 is rabbits per week, and t is time in weeks. Below is a graph of the curves representing the rates of growth of the two populations over the observation period: 10 20 30 50 60 70 80 A. Using your calculator, find (approximately) the first positive time t for which the rates of growth for the two populations are the same. B. What's the physical significance of the area between the two curves from time t = 0 until the first time where the two rates are the same? What does the area represent? C. Suppose you want to find the first time (call it 7) after the beginning of the observation period at which the two rabbit populations…arrow_forwardProblem 9: Suppose that y = y(t) is a twice differentiable function of exponential type and that y"(t) + w?y(t) = where w is any real number. Show that 1 (L{y(t)})(s) = y(0)- + y/ (0)- s2 + w2 s2 + w2arrow_forwardProblem 8 If ä – i – 6x = 0 with x(0) = 1 and i(0) = a, determine the value(s) of a such that the solution tends to zero at t 0.arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- Discrete Mathematics and Its Applications ( 8th I...MathISBN:9781259676512Author:Kenneth H RosenPublisher:McGraw-Hill EducationMathematics for Elementary Teachers with Activiti...MathISBN:9780134392790Author:Beckmann, SybillaPublisher:PEARSON
- Thinking Mathematically (7th Edition)MathISBN:9780134683713Author:Robert F. BlitzerPublisher:PEARSONDiscrete Mathematics With ApplicationsMathISBN:9781337694193Author:EPP, Susanna S.Publisher:Cengage Learning,Pathways To Math Literacy (looseleaf)MathISBN:9781259985607Author:David Sobecki Professor, Brian A. MercerPublisher:McGraw-Hill Education
Discrete Mathematics and Its Applications ( 8th I...
Math
ISBN:9781259676512
Author:Kenneth H Rosen
Publisher:McGraw-Hill Education
Mathematics for Elementary Teachers with Activiti...
Math
ISBN:9780134392790
Author:Beckmann, Sybilla
Publisher:PEARSON
Thinking Mathematically (7th Edition)
Math
ISBN:9780134683713
Author:Robert F. Blitzer
Publisher:PEARSON
Discrete Mathematics With Applications
Math
ISBN:9781337694193
Author:EPP, Susanna S.
Publisher:Cengage Learning,
Pathways To Math Literacy (looseleaf)
Math
ISBN:9781259985607
Author:David Sobecki Professor, Brian A. Mercer
Publisher:McGraw-Hill Education
Intro to the Laplace Transform & Three Examples; Author: Dr. Trefor Bazett;https://www.youtube.com/watch?v=KqokoYr_h1A;License: Standard YouTube License, CC-BY