Fundamentals of Differential Equations and Boundary Value Problems
7th Edition
ISBN: 9780321977106
Author: Nagle, R. Kent
Publisher: Pearson Education, Limited
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Textbook Question
Chapter 4.3, Problem 1E
In Problems 1-8, the auxiliary equation for the given differential equation has complex roots. Find the general solution.
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Exercises
Evaluate the following limits.
1. lim cot x/ln x
+01x
2. lim x² In x
+014
3. lim x*
x0+
4. lim (cos√√x)1/x
+014
5. lim x2/(1-cos x)
x10
6. lim e*/*
818
7. lim (secx - tan x)
x-x/2-
8. lim [1+(3/x)]*
x→∞0
In Exercises 1 through 3, let xo =
O and calculate P7(x) and R7(x).
1. f(x)=sin x, x in R.
2. f(x) = cos x, x in R.
3. f(x) = In(1+x), x≥0.
4. In Exercises 1, 2, and 3, for |x| 1, calculate a value of n such that P(x)
approximates f(x) to within 10-6.
5. Let (an)neN be a sequence of positive real numbers such that L =
lim (an+1/an) exists in R. If L < 1, show that an → 0. [Hint: Let
1111
L
iation
7. Let f be continuous on [a, b] and differentiable on (a, b). If lim f'(x)
xia
exists in R, show that f is differentiable at a and f'(a) = lim f'(x). A
similar result holds for b.
x-a
8. In reference to Corollary 5.4, give an example of a uniformly continuous
function on [0, 1] that is differentiable on (0, 1] but whose derivative is not
bounded there.
9. Recall that a fixed point of a function f is a point c such that f(c) = c.
(a) Show that if f is differentiable on R and f'(x)| x if x 1 and hence In(1+x) 0.
12. For 0 л/2. (Thus,
as x л/2 from the left, cos x is never large enough for x+cosx to be
greater than л/2 and cot x is never small enough for x + cot x to be less
than x/2.)
Chapter 4 Solutions
Fundamentals of Differential Equations and Boundary Value Problems
Ch. 4.1 - Verify that for b=0 and Fext(t)=0, equation (3)...Ch. 4.1 - If Fext(t)=0, equation (3) becomes my+by+ky=0. For...Ch. 4.1 - Show that if Fext(t)=0, m=1, k=9, and b=6, then...Ch. 4.1 - Prob. 4ECh. 4.1 - Verify that the exponentially damped sinusoid...Ch. 4.1 - An external force F(t)=2cos2t is applied to a...Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - Undamped oscillators that are driven at resonance...
Ch. 4.2 - In Problems 1-12, find a general solution to the...Ch. 4.2 - In Problems 1-12, find a general solution to the...Ch. 4.2 - In Problems 1-12, find a general solution to the...Ch. 4.2 - In Problems 1-12, find a general solution to the...Ch. 4.2 - In Problems 1-12, find a general solution to the...Ch. 4.2 - In Problems 1-12, find a general solution to the...Ch. 4.2 - In Problems 1-12, find a general solution to the...Ch. 4.2 - In Problems 1-12, find a general solution to the...Ch. 4.2 - In Problems 1-12, find a general solution to the...Ch. 4.2 - In Problems 1-12, find a general solution to the...Ch. 4.2 - In Problems 1-12, find a general solution to the...Ch. 4.2 - In Problems 1-12, find a general solution to the...Ch. 4.2 - In Problems 13-20, solve the given initial value...Ch. 4.2 - In Problems 13-20, solve the given initial value...Ch. 4.2 - In Problems 13-20, solve the given initial value...Ch. 4.2 - In Problems 13-20, solve the given initial value...Ch. 4.2 - In Problems 13-20, solve the given initial value...Ch. 4.2 - In Problems 13-20, solve the given initial value...Ch. 4.2 - In Problems 13-20, solve the given initial value...Ch. 4.2 - Prob. 20ECh. 4.2 - Prob. 21ECh. 4.2 - Prob. 22ECh. 4.2 - In Problems 22-25, use the method described in...Ch. 4.2 - Prob. 24ECh. 4.2 - In Problems 22-25, use the method described in...Ch. 4.2 - 26.Boundary Value Problems. When the values of a...Ch. 4.2 - In Problems 27 32, use Definition 1 to determine...Ch. 4.2 - In Problems 2732, use Definition 1 to determine...Ch. 4.2 - Prob. 29ECh. 4.2 - Prob. 30ECh. 4.2 - In Problems 2732, use Definition 1 to determine...Ch. 4.2 - Prob. 32ECh. 4.2 - Prob. 33ECh. 4.2 - Wronskian. For any two differentiable functions y1...Ch. 4.2 - Prob. 35ECh. 4.2 - Prob. 36ECh. 4.2 - In Problems 3741, find three linearly independent...Ch. 4.2 - Prob. 38ECh. 4.2 - In Problems 3741, find three linearly independent...Ch. 4.2 - In Problems 3741, find three linearly independent...Ch. 4.2 - Prob. 41ECh. 4.2 - Prob. 42ECh. 4.2 - Prob. 43ECh. 4.2 - Solve the initial value problem: y2yy+2y=0;...Ch. 4.2 - Prob. 45ECh. 4.2 - Prob. 46ECh. 4.3 - In Problems 1-8, the auxiliary equation for the...Ch. 4.3 - In Problems 1-8, the auxiliary equation for the...Ch. 4.3 - Prob. 3ECh. 4.3 - Prob. 4ECh. 4.3 - In Problems 1-8, the auxiliary equation for the...Ch. 4.3 - In Problems 1-8, the auxiliary equation for the...Ch. 4.3 - In Problems 1-8, the auxiliary equation for the...Ch. 4.3 - In Problems 1-8, the auxiliary equation for the...Ch. 4.3 - In Problems 9-20, find a general solution....Ch. 4.3 - In Problems 9-20, find a general solution....Ch. 4.3 - In Problems 9-20, find a general solution....Ch. 4.3 - In Problems 9-20, find a general solution. u+7u=0Ch. 4.3 - In Problems 9-20, find a general solution....Ch. 4.3 - In Problems 9-20, find a general solution....Ch. 4.3 - In Problems 9-20, find a general solution....Ch. 4.3 - In Problems 9-20, find a general solution....Ch. 4.3 - In Problems 9-20, find a general solution. yy+7y=0Ch. 4.3 - In Problems 9-20, find a general solution....Ch. 4.3 - In Problems 9-20, find a general solution....Ch. 4.3 - In Problems 9-20, find a general solution. yy+2y=0Ch. 4.3 - In Problems 21-27, solve the given initial value...Ch. 4.3 - In Problems 21-27, solve the initial value...Ch. 4.3 - In Problems 21-27, solve the given initial value...Ch. 4.3 - In Problems 21-27, solve the given initial value...Ch. 4.3 - In Problems 21-27, solve the given initial value...Ch. 4.3 - Prob. 26ECh. 4.3 - In Problems 21-27, solve the given initial value...Ch. 4.3 - To see the effect of changing the parameters b in...Ch. 4.3 - Find a general solution to the following...Ch. 4.3 - Prob. 30ECh. 4.3 - Using the mass-spring analogy, predict the...Ch. 4.3 - Vibrating Spring without Damping. A vibrating...Ch. 4.3 - Vibrating Spring with Damping. Using the model for...Ch. 4.3 - Prob. 34ECh. 4.3 - Swinging Door. The motion of a swinging door with...Ch. 4.3 - Prob. 36ECh. 4.3 - Prob. 37ECh. 4.3 - Prove the sum of angles formula for the sine...Ch. 4.4 - In Problem 1-8, decide whether or not the method...Ch. 4.4 - In Problem 1-8, decide whether or not the method...Ch. 4.4 - In Problem 1-8, decide whether or not the method...Ch. 4.4 - In Problem 1-8, decide whether or not the method...Ch. 4.4 - In Problem 1-8, decide whether or not the method...Ch. 4.4 - Prob. 6ECh. 4.4 - In Problem 1-8, decide whether or not the method...Ch. 4.4 - Prob. 8ECh. 4.4 - In Problems 9-26, find a particular solution to...Ch. 4.4 - In Problems 9-26, find a particular solution to...Ch. 4.4 - In Problem 9-26, find a particular solution to the...Ch. 4.4 - Prob. 12ECh. 4.4 - In Problems 9-26, find a particular solution to...Ch. 4.4 - In Problems 9-26, find a particular solution to...Ch. 4.4 - In Problems 9-26, find a particular solution to...Ch. 4.4 - Prob. 16ECh. 4.4 - In Problems 9-26, find a particular solution to...Ch. 4.4 - Prob. 18ECh. 4.4 - In Problems 9-26, find a particular solution to...Ch. 4.4 - Prob. 20ECh. 4.4 - In Problems 9-26, find a particular solution to...Ch. 4.4 - In Problems 9-26, find a particular solution to...Ch. 4.4 - In Problems 9-26, find a particular solution to...Ch. 4.4 - In Problems 9-26, find a particular solution to...Ch. 4.4 - In Problems 9-26, find a particular solution to...Ch. 4.4 - In Problems 9-26, find a particular solution to...Ch. 4.4 - In Problems 2732, determine the form of a...Ch. 4.4 - In Problems 27 32, determine the form of a...Ch. 4.4 - In Problems 2732, determine the form of a...Ch. 4.4 - In Problems 2732, determine the form of a...Ch. 4.4 - Prob. 31ECh. 4.4 - In Problems 2732, determine the form of a...Ch. 4.4 - Prob. 33ECh. 4.4 - In Problems 3336, use the method of undetermined...Ch. 4.4 - Prob. 35ECh. 4.4 - In Problems 3336, use the method of undetermined...Ch. 4.5 - Given that y1(t)=cost is a solution to yy+y=sint...Ch. 4.5 - Given that y1(t)=(1/4)sin2t is a solution to...Ch. 4.5 - In Problems 3-8, a nonhomogeneous equation and a...Ch. 4.5 - In Problem 3-8, a nonhomogeneous equation and a...Ch. 4.5 - In Problem 3-8, a nonhomogeneous equation and a...Ch. 4.5 - In Problems 3-8, a nonhomogeneous equation and a...Ch. 4.5 - In Problems 3-8, a nonhomogeneous equation and a...Ch. 4.5 - In Problems 3-8, a nonhomogeneous equation and a...Ch. 4.5 - Prob. 9ECh. 4.5 - Prob. 10ECh. 4.5 - Prob. 11ECh. 4.5 - In Problems 9-16 decide whether the method of...Ch. 4.5 - In Problems 9-16 decide whether the method of...Ch. 4.5 - Prob. 14ECh. 4.5 - Prob. 15ECh. 4.5 - Prob. 16ECh. 4.5 - Prob. 17ECh. 4.5 - In Problem 17-22, find a general solution to the...Ch. 4.5 - In Problems 17-22, find a general solution to the...Ch. 4.5 - In Problems 17-22, find a general solution to the...Ch. 4.5 - Prob. 21ECh. 4.5 - Prob. 22ECh. 4.5 - Prob. 23ECh. 4.5 - Prob. 24ECh. 4.5 - In Problems 2330, find the solution to the initial...Ch. 4.5 - In Problems 2330, find the solution to the initial...Ch. 4.5 - Prob. 27ECh. 4.5 - In Problems 2330, find the solution to the initial...Ch. 4.5 - Prob. 29ECh. 4.5 - Prob. 30ECh. 4.5 - Prob. 31ECh. 4.5 - In Problems 3136, determine the form of a...Ch. 4.5 - In Problems 3136, determine the form of a...Ch. 4.5 - Prob. 34ECh. 4.5 - In Problems 3136, determine the form of a...Ch. 4.5 - In Problems 31 36, determine the form of a...Ch. 4.5 - In Problems 3740, find a particular solution to...Ch. 4.5 - Prob. 38ECh. 4.5 - Prob. 39ECh. 4.5 - Prob. 40ECh. 4.5 - Discontinuous Forcing Term. In certain physical...Ch. 4.5 - Forced Vibrations. As discussed in Section 4.1, a...Ch. 4.5 - A massspring system is driven by a sinusoidal...Ch. 4.5 - Prob. 44ECh. 4.5 - Speed Bumps. Often bumps like the one depicted in...Ch. 4.5 - Prob. 46ECh. 4.5 - Prob. 47ECh. 4.5 - Prob. 48ECh. 4.6 - In Problems 18, find a general solution to the...Ch. 4.6 - In Problems 18, find a general solution to the...Ch. 4.6 - Prob. 3ECh. 4.6 - In Problems 18, find a general solution to the...Ch. 4.6 - In Problems 18, find a general solution to the...Ch. 4.6 - In Problems 18, find a general solution to the...Ch. 4.6 - In Problems 18, find a general solution to the...Ch. 4.6 - In Problems 18, find a general solution to the...Ch. 4.6 - Prob. 9ECh. 4.6 - In Problems 9 and 10, find a particular solution...Ch. 4.6 - In Problems 1118, find a general solution to the...Ch. 4.6 - In Problems 1118, find a general solution to the...Ch. 4.6 - Prob. 13ECh. 4.6 - In Problems 11-18, find a general solution to the...Ch. 4.6 - In Problems 11-18, find a general solution to the...Ch. 4.6 - In Problems 11-18, find a general solution to the...Ch. 4.6 - In Problems 11-18, find a general solution to the...Ch. 4.6 - In Problems 11-18, find a general solution to the...Ch. 4.6 - Prob. 19ECh. 4.6 - Use the method of variation of parameters to show...Ch. 4.6 - Prob. 21ECh. 4.6 - Prob. 22ECh. 4.6 - Prob. 23ECh. 4.6 - In Problems 22 through 25, use variation of...Ch. 4.6 - In Problems 22 through 25, use variation of...Ch. 4.7 - In Problems 1 through 4, use Theorem 5 to discuss...Ch. 4.7 - In Problems 1 through 4, use Theorem 5 to discuss...Ch. 4.7 - In Problems 1 through 4, use Theorem 5 to discuss...Ch. 4.7 - In Problems 1 through 4, use Theorem 5 to discuss...Ch. 4.7 - In Problems 5 through 8, determine whether Theorem...Ch. 4.7 - In Problems 5 through 8, determine whether Theorem...Ch. 4.7 - In Problems 5 through 8, determine whether Theorem...Ch. 4.7 - In Problems 5 through 8, determine whether Theorem...Ch. 4.7 - In Problems 9 through 14, find a general solution...Ch. 4.7 - Prob. 10ECh. 4.7 - Prob. 11ECh. 4.7 - Prob. 12ECh. 4.7 - In Problems 9 through 14, find a general solution...Ch. 4.7 - Prob. 14ECh. 4.7 - Prob. 15ECh. 4.7 - In Problems 15 through 18, find a general solution...Ch. 4.7 - In Problems 15 through 18, find a general solution...Ch. 4.7 - In Problems 15 through 18, find a general solution...Ch. 4.7 - Prob. 19ECh. 4.7 - Prob. 20ECh. 4.7 - Prob. 21ECh. 4.7 - In Problems 21 and 22, devise a modification of...Ch. 4.7 - Prob. 23ECh. 4.7 - Prob. 24ECh. 4.7 - Prob. 25ECh. 4.7 - Let y1(t)=t3 and y2(t)=|t3|. Are y1 and y2...Ch. 4.7 - Prob. 27ECh. 4.7 - Let y1(t)=t2 and y2(t)=2t|t|. Are y1 and y2...Ch. 4.7 - Prob. 29ECh. 4.7 - Prob. 30ECh. 4.7 - Prob. 31ECh. 4.7 - By completing the following steps, prove that the...Ch. 4.7 - Prob. 33ECh. 4.7 - Given that 1+t, 1+2t, and 1+3t2 are solutions to...Ch. 4.7 - Verify that the given functions y1 and y2 are...Ch. 4.7 - In Problems 37 through 39, find general solutions...Ch. 4.7 - Prob. 38ECh. 4.7 - In Problems 37 through 39, find general solutions...Ch. 4.7 - Prob. 40ECh. 4.7 - In Problems 41 through 44, a differential equation...Ch. 4.7 - In Problems 41 through 44, a differential equation...Ch. 4.7 - In Problems 41 through 44, a differential equation...Ch. 4.7 - In Problems 41 through 44, a differential equation...Ch. 4.7 - Find a particular solution to the nonhomogeneous...Ch. 4.7 - Find a particular solution to the nonhomogeneous...Ch. 4.7 - In quantum mechanics, the study of the Schrodinger...Ch. 4.7 - Prob. 48ECh. 4.7 - Prob. 49ECh. 4.7 - Prob. 50ECh. 4.7 - Prob. 51ECh. 4.7 - Prob. 52ECh. 4.8 - Show that if y(t) satisfies yty=0, then y(t)...Ch. 4.8 - Prob. 2ECh. 4.8 - Prob. 3ECh. 4.8 - Prob. 4ECh. 4.8 - a. Use the energy integral lemma to derive the...Ch. 4.8 - Prob. 6ECh. 4.8 - Prob. 7ECh. 4.8 - Use the energy integral Lemma to show that...Ch. 4.8 - Prob. 9ECh. 4.8 - Prob. 10ECh. 4.8 - Prob. 11ECh. 4.8 - Prob. 12ECh. 4.8 - Prob. 13ECh. 4.8 - Prob. 14ECh. 4.8 - Use the mass-spring oscillator analogy to decide...Ch. 4.8 - Prob. 16ECh. 4.8 - Prob. 17ECh. 4.9 - All problems refer to the mass-spring...Ch. 4.9 - Prob. 2ECh. 4.9 - All problems refer to the mass-spring...Ch. 4.9 - All problems refer to the mass-spring...Ch. 4.9 - Prob. 5ECh. 4.9 - Prob. 6ECh. 4.9 - Prob. 7ECh. 4.9 - Prob. 8ECh. 4.9 - A 2kg mass is attached to a spring with stiffness...Ch. 4.9 - A 1/4-kg mass is attached to a spring with...Ch. 4.9 - Prob. 11ECh. 4.9 - A 1/4-kg mass is attached to a spring with...Ch. 4.9 - Prob. 13ECh. 4.9 - For an underdamped system, verify that as b0 the...Ch. 4.9 - How can one deduce the value of the damping...Ch. 4.9 - Prob. 16ECh. 4.9 - Consider the equation for free mechanical...Ch. 4.9 - Consider the equation for free mechanical...Ch. 4.10 - Sketch the frequency response curve (13) for the...Ch. 4.10 - Prob. 2ECh. 4.10 - Determine the equation of the motion for an...Ch. 4.10 - Prob. 4ECh. 4.10 - An undamped system is governed by...Ch. 4.10 - Derive the formula for yp(t) given in 21...Ch. 4.10 - Shock absorbers in automobiles and aircraft can be...Ch. 4.10 - The response of an overdamped system to a constant...Ch. 4.10 - An 8-kg mass is attached to a spring hanging from...Ch. 4.10 - Show that the period of the simple harmonic motion...Ch. 4.10 - A mass weighing 8 lb is attached to a spring...Ch. 4.10 - A 2-kg mass is attached to a spring hanging from...Ch. 4.10 - A mass weighing 32lb is attached to a spring...Ch. 4.10 - An 8-kg mass is attached to a spring hanging from...Ch. 4.10 - An 8-kg mass is attached to a spring hanging from...Ch. 4.RP - In Problems 1-28, find a general solution to the...Ch. 4.RP - Prob. 2RPCh. 4.RP - Prob. 3RPCh. 4.RP - Prob. 4RPCh. 4.RP - Prob. 5RPCh. 4.RP - Prob. 6RPCh. 4.RP - Prob. 7RPCh. 4.RP - Prob. 8RPCh. 4.RP - In Problems 1 -28, find the general solution to...Ch. 4.RP - Prob. 10RPCh. 4.RP - Prob. 11RPCh. 4.RP - Prob. 12RPCh. 4.RP - Prob. 13RPCh. 4.RP - Prob. 14RPCh. 4.RP - Prob. 15RPCh. 4.RP - Prob. 16RPCh. 4.RP - Prob. 17RPCh. 4.RP - Prob. 18RPCh. 4.RP - Prob. 19RPCh. 4.RP - Prob. 20RPCh. 4.RP - Prob. 21RPCh. 4.RP - Prob. 22RPCh. 4.RP - Prob. 23RPCh. 4.RP - Prob. 24RPCh. 4.RP - Prob. 25RPCh. 4.RP - In Problems 1-28, find a general solution to the...Ch. 4.RP - Prob. 27RPCh. 4.RP - Prob. 28RPCh. 4.RP - Prob. 29RPCh. 4.RP - Prob. 30RPCh. 4.RP - Prob. 31RPCh. 4.RP - Prob. 32RPCh. 4.RP - Prob. 33RPCh. 4.RP - Prob. 34RPCh. 4.RP - Prob. 35RPCh. 4.RP - Prob. 36RPCh. 4.RP - Use the mass-spring oscillator analogy to decide...Ch. 4.RP - A 3kg mass is attached to a spring with stiffness...Ch. 4.RP - A 32lb weight is attached to a vertical spring,...
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Publisher:McGraw-Hill Education
01 - What Is A Differential Equation in Calculus? Learn to Solve Ordinary Differential Equations.; Author: Math and Science;https://www.youtube.com/watch?v=K80YEHQpx9g;License: Standard YouTube License, CC-BY
Higher Order Differential Equation with constant coefficient (GATE) (Part 1) l GATE 2018; Author: GATE Lectures by Dishank;https://www.youtube.com/watch?v=ODxP7BbqAjA;License: Standard YouTube License, CC-BY
Solution of Differential Equations and Initial Value Problems; Author: Jefril Amboy;https://www.youtube.com/watch?v=Q68sk7XS-dc;License: Standard YouTube License, CC-BY