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mathematics
a first course in differential equations

A First Course in Differential Equations with Modeling Applications 10th edition Dennis G. Zill - Solutions

In Problem 31 determine the equation of motion if the external force is f(t) = e-t sin 4t. Analyze the displacements for t → ∞.
The model mx'' + kx = 0 for simple harmonic motion, discussed in Section 5.1, can be related to Example 2 of this section.Consider a free undamped spring/mass system for which the spring constant is, say, k = 10 lb/ft. Determine those masses mn that can be attached to the spring so that when each
Consider a pendulum that is released from rest from an initial displacement of θ0 radians. Solving the linear model (7) subject to the initial conditions θ(0) = θ0, θ'(0) = 0 gives θ(t) = θ0 cos √g/lt. The period of oscillations predicted by this model is given by the familiar formula Y =
After a mass weighing 10 pounds is attached to a 5-foot spring, the spring measures 7 feet. This mass is removed and replaced with another mass that weighs 8 pounds. The entire system is placed in a medium that offers a damping force that is numerically equal to the instantaneous velocity.(a) Find
A force of 2 pounds stretches a spring 1 foot. A mass weighing 3.2 pounds is attached to the spring, and the system is then immersed in a medium that offers a damping force that is numerically equal to 0.4 times the instantaneous velocity.(a) Find the equation of motion if the mass is initially
A bead is constrained to slide along a frictionless rod of length L. The rod is rotating in a vertical plane with a constant angular velocity ω about a pivot P fixed at the midpoint of the rod, but the design of the pivot allows the bead to move along the entire length of the rod. Let r(t) denote
As shown in Figure 5.3.11, a plane flying horizontally at a constant speed v0 drops a relief supply pack to a person on the ground. Assume the origin is the point where the supply pack is released and that the positive x-axis points forward and that positive y-axis points downward. Under the $dx\2 i+ dv m dt mg - k dt j$
Historically, in order to maintain quality control over munitions (bullets) produced by an assembly line, the manufacturer would use a ballistic pendulum to determine the muzzle velocity of a gun, that is, the speed of a bullet as it leaves the barrel. Invented in 1742 by the English engineer
A mass weighing 10 pounds stretches a spring 1/3 foot. This mass is removed and replaced with a mass of 1.6 slugs, which is initially released from a point 1/3 foot above the equilibrium position with a downward velocity of ¾ ft/s.(a) Express the equation of motion in the form given in (6).(b)
In problem proceed as in Example 3 to rewrite the given expression using a single power series whose general term involves xk.
Use an appropriate series in (2) to find the Maclaurin series of the given function. Write your answer in summation notation.1/2 + x
Use an appropriate series in (2) to find the Maclaurin series of the given function. Write your answer in summation notation.xe3x
Use an appropriate series in (2) to find the Maclaurin series of the given function. Write your answer in summation notation.e-x/2
Find the interval and radius of convergence for the given power series. 23-(4x - 5)* k=0
Find the interval and radius of convergence for the given power series. (3x 1) k=1 k² + k
Find the interval and radius of convergence for the given power series. Ek!(x E k!(x – 1)* k=0
Find the interval and radius of convergence for the given power series. (-1)시 (x- 5)* k=1 10%
Find the interval and radius of convergence for the given power series. in Σ n=on!
Find the interval and radius of convergence for the given power series. on
Find the interval and radius of convergence for the given power series. (-1)" n=0 9"
Find the interval and radius of convergence for the given power series. 25k Σ 52k3, k k=1 8
Find the interval and radius of convergence for the given power series. ent
Find the interval and radius of convergence for the given power series.
Consider the boundary-value problemy'' + y = 0, y(0) = y(2π), y'(0) = y'(2π)Show that except for the case λ = 0, there are two independent eigenfunctions corresponding to each eigenvalue.
(a) Show that the current i(t) in an LRC-series circuit satisfieswhere E'(t) denotes the derivative of E(t).(b) Two initial conditions i(0) and i'(0) can be specified for the DE in part (a). If i(0) = i0 and q(0) = q0, what is i'(0)? d?i di = E'(1), C L + R df dt
A series circuit contains an inductance of L = 1 h, a capacitance of C = 10-4 f, and an electromotive force of E(t) = 100 sin 50t V. Initially, the charge q and current i are zero.(a) Determine the charge q(t).(b) Determine the current i(t).(c) Find the times for which the charge on the capacitor
(a) Two springs are attached in series as shown in Figure 5.R.1. If the mass of each spring is ignored, show that the effective spring constant k of the system is defined by 1/k = 1/k1 + 1/k2.(b) A mass weighing W pounds stretches a spring foot and stretches a different spring foot. The two springs
A mass weighing 4 pounds is suspended from a spring whose constant is 3 lb/ft. The entire system is immersed in a fluid offering a damping force numerically equal to the instantaneous velocity. Beginning at t = 0, an external force equal to f(t) = e-t is impressed on the system. Determine the
Find a particular solution for x'' + 2λx' + ω2x = A, where A is a constant force.
A mass weighing 4 pounds stretches a spring 18 inches. A periodic force equal to f(t) = cos γt + sin γt is impressed on the system starting at t = 0. In the absence of a damping force, for what value of g will the system be in a state of pure resonance?
The vertical motion of a mass attached to a spring is described by the IVP 1/4x'' + x' + x = 0, x(0) = 4, x'(0) = 2. Determine the maximum vertical displacement of the mass.
A spring with constant k = 2 is suspended in a liquid that offers a damping force numerically equal to 4 times the instantaneous velocity. If a mass m is suspended from the spring, determine the values of m for which the subsequent free motion is nonoscillatory.
A mass weighing 32 pounds stretches a spring 6 inches. The mass moves through a medium offering a damping force that is numerically equal to b times the instantaneous velocity. Determine the values of β > 0 for which the spring/mass system will exhibit oscillatory motion.
A force of 2 pounds stretches a spring 1 foot. With one end held fixed, a mass weighing 8 pounds is attached to the other end. The system lies on a table that imparts a frictional force numerically equal to 3/2 times the instantaneous velocity. Initially, the mass is displaced 4 inches above the
Amass weighing 12 pounds stretches a spring 2 feet. The mass is initially released from a point 1 foot below the equilibrium position with an upward velocity of 4 ft/s.(a) Find the equation of motion.(b) What are the amplitude, period, and frequency of the simple harmonic motion?(c) At what times
A free undamped spring/mass system oscillates with a period of 3 seconds. When 8 pounds are removed from the spring, the system has a period of 2 seconds. What was the weight of the original mass on the spring?
In problem the eigenvalues and eigenfunctions of the boundary-value problem y'' + λy = 0, y'(0) = 0, y'(π) = 0 are λn = n2, n = 0, 1, 2, . . . , and y = cos nx, respectively. Fill in the blanks.A solution of the BVP when λ = 36 is y = ______ because ____.
In problem the eigenvalues and eigenfunctions of the boundary-value problem y'' + λy = 0, y'(0) = 0, y'(π) = 0 are λn = n2, n = 0, 1, 2, . . . , and y = cos nx, respectively. Fill in the blanks.A solution of the BVP when λ = 8 is y = ____ because _____.
If simple harmonic motion is described by x = (√2/2)sin(2t + ϕ), the phase angle ϕ is ______ when the initial conditions x(0) = -1/2 and x'(0) = 1.Answer problem without referring back to the text. Fill in the blank or answer true/false.
At critical damping any increase in damping will result in an ______ system.Answer problem without referring back to the text. Fill in the blank or answer true/false.
A mass on a spring whose motion is critically damped can possibly pass through the equilibrium position twice. _______Answer problem without referring back to the text. Fill in the blank or answer true/false.
In the presence of a damping force, the displacements of a mass on a spring will always approach zero as t → ∞, _____Answer problem without referring back to the text. Fill in the blank or answer true/false.
Pure resonance cannot take place in the presence of a damping force. ______Answer problem without referring back to the text. Fill in the blank or answer true/false.
The differential equation of a spring/mass system is x'' + 16x = 0. If the mass is initially released from a point 1 meter above the equilibrium position with a downward velocity of 3 m/s, the amplitude of vibrations is _______ meters.Answer problem without referring back to the text. Fill in the
The period of simple harmonic motion of mass weighing 8 pounds attached to a spring whose constant is 6.25 lb/ft is _______ seconds.Answer problem without referring back to the text. Fill in the blank or answer true/false.
If a mass weighing 10 pounds stretches a spring 2.5 feet, a mass weighing 32 pounds will stretch it ______ feet.Fill in the blank.
Consider the initial-value problemfor a nonlinear pendulum. Since we cannot solve the differential equation, we can find no explicit solution of this problem. But suppose we wish to determine the first time t1 > 0 for which the pendulum in Figure 5.3.3, starting from its initial position to the
Repeat the two parts of Problem 23 this time using the linear model (7).Problem 23Does a pendulum of length l oscillate faster on the Earth or on the Moon?(a) Take l = 3 and g = 32 for the acceleration of gravity on Earth. Use a numerical solver to generate a numerical solution curve for the
Does a pendulum of length l oscillate faster on the Earth or on the Moon?(a) Take l = 3 and g = 32 for the acceleration of gravity on Earth. Use a numerical solver to generate a numerical solution curve for the nonlinear model (6) subject to the initial conditions θ(0) = 1, θ'(0) = 2. Repeat
(a) Experiment with a calculator to find an interval 0 ≤ θ < u1, where θ is measured in radians, for which you think sin θ ≈ θ is a fairly good estimate. Then use a graphing utility to plot the graphs of y = x and y = sin x on the same coordinate axes for 0 ≤ x ≤ π/2. Do the graphs
Discuss why the damping term in equation (3) is written as dx|dx B dt dt instead of B dt
In another naval exercise a destroyer S1 pursues a submerged submarine S2. Suppose that S1 at (9, 0) on the x-axis detects S2 at (0, 0) and that S2 simultaneously detects S1. The captain of the destroyer S1 assumes that the submarine will take immediate evasive action and conjectures that its
In a naval exercise a ship S1 is pursued by a submarine S2 as shown in Figure 5.3.8. Ship S1 departs point (0, 0) at t = 0 and proceeds along a straightline course (the y-axis) at a constant speed v1. The submarine S2 keeps ship S1 in visual contact, indicated by the straight dashed line L in
A uniform chain of length L, measured in feet, is held vertically so that the lower end just touches the floor. The chain weighs 2 lb/ft. The upper end that is held is released from rest at t = 0 and the chain falls straight down. If x(t) denotes the length of the chain on the floor at time t, air
(a) In Example 4, how much of the chain would you intuitively expect the constant 5-pound force to be able to lift?(b) What is the initial velocity of the chain?(c) Why is the time interval corresponding to x(t) « 0 given in Figure 5.3.7 not the interval I of definition of the solution
(a) Use the substitution v = dy/dt to solve (13) for v in terms of y. Assuming that the velocity of the rocket at burnout is v = v0 and y ≈ R at that instant, show that the approximate value of the constant c of integration is c = -gR + ½ v20.(b) Use the solution for v in part (a) to show that
Consider the model of the free damped nonlinear pendulum given byUse a numerical solver to investigate whether the motion in the two cases λ2 – ω2 0 and λ2-ω2 < 0 corresponds, respectively, to the overdamped and underdamped cases discussed in Section 5.1 for spring/mass
(a) Find values of k1 < 0 for which the system in Problem 11 is oscillatory.(b) Consider the initial-value problemx'' = x + k1x3 = , x(0) = 0, x'(0) = 0.Find values for k1 < 0 for which the system is oscillatory.Problem 11The model mx'' + kx + k1x3 = F0cos ωt of an undamped periodically
The model mx'' + kx + k1x3 = F0cos ωt of an undamped periodically driven spring/mass system is called Duffing s differential equation. Consider the initial-value problem x'' + x + k1x3 = 5 cos t, x(0) = 1, x'(0) = 0. Use a numerical solver to investigate the behavior of the system for values of k1
In problem the given differential equation is a model of a damped nonlinear spring/mass system. Predict the behavior of each system as t → ∞. For each equation use a numerical solver to obtain the solution curves satisfying the given initial conditions. dx + x - x = 0, dt d?x dr %3D x(0) = 0,
In problem the given differential equation is a model of a damped nonlinear spring/mass system. Predict the behavior of each system as t → ∞. For each equation use a numerical solver to obtain the solution curves satisfying the given initial conditions. d²x, dx + x + x = 0, dt di? x(0) = -3,
Consider the model of an undamped nonlinear spring/mass system given by x'' + 8x - 6x3 + x5 = 0.Use a numerical solver to discuss the nature of the oscillations of the system corresponding to the initial conditions: x(0) = -2, x'(0) = x(0) = V2, x'(0) = 1; x(0) = 2, x'(0) = : x(0) = 1, x'(0) = 1;
Find a linearization of the differential equation in Problem 4.Problem 4.In problem the given differential equation is model of an undamped spring/mass system in which the restoring force F(x) in (1) is nonlinear. For each equation use a numerical solver to plot the solution curves that satisfy the
In Problem 3, suppose the mass is released from an initial position x(0) = x0 with the initial velocity x'(0) = 1. Use a numerical solver to estimate an interval a ≤ x0 ≤ b for which the motion is oscillatory.Problem 3In problem the given differential equation is model of an undamped
In Problem 3, suppose the mass is released from the initial position x(0) = 1 with an initial velocity x'(0) = x1. Use a numerical solver to estimate the smallest value of x1 at which the motion of the mass is nonperiodic.Problem 3In problem the given differential equation is model of an undamped
In problem the given differential equation is model of an undamped spring/mass system in which the restoring force F(x) in (1) is nonlinear. For each equation use a numerical solver to plot the solution curves that satisfy the given initial conditions. If the solutions appear to be periodic use the
In problem the given differential equation is model of an undamped spring/mass system in which the restoring force F(x) in (1) is nonlinear. For each equation use a numerical solver to plot the solution curves that satisfy the given initial conditions. If the solutions appear to be periodic use the
In problem the given differential equation is model of an undamped spring/mass system in which the restoring force F(x) in (1) is nonlinear. For each equation use a numerical solver to plot the solution curves that satisfy the given initial conditions. If the solutions appear to be periodic use the
In problem the given differential equation is model of an undamped spring/mass system in which the restoring force F(x) in (1) is nonlinear. For each equation use a numerical solver to plot the solution curves that satisfy the given initial conditions. If the solutions appear to be periodic use the
Find the eigenvalues and eigenfunctions of the given boundary-value problem. Use a CAS to approximate the first four eigenvalues λ1, λ2, λ3, and λ4.y(4) – λy = 0, y(0) = 0, y'(0) = 0, y(1) = 0, y'(1) = 0Consider only λ = α4, α > 0.
Find the eigenvalues and eigenfunctions of the given boundary-value problem. Use a CAS to approximate the first four eigenvalues λ1, λ2, λ3, and λ4.y'' + y = 0, y(0) = 0, y(1) - 1/2y'(1) = 0
Use a root-finding application of a CAS to approximate the first four eigenvalues λ1, λ2, λ3, and λ4 for the BVP in Problem 34.
Use a CAS to plot graphs to convince yourself that the equation tan α = -α in Problem 34 has an infinite number of roots. Explain why the negative roots of the equation can be ignored. Explain why λ = 0 is not an eigenvalue even though α = 0 is an obvious solution of the equation tan α = -α.
Show that the eigenvalues and eigenfunctions of the boundary-value problemy'' + y = 0, y(0) = 0, y(1) + y'(1) = 0are n = α2n yn = sin αn x, respectively, where an, n = 1, 2, 3, . . . are the consecutive positive roots of the equation tan α = -α.
Consider the boundary-value problemy'' + y = 0, y(-π) = y(π), y'(-π) = y'(π).(a) The type of boundary conditions specified are called periodic boundary conditions. Give a geometric interpretation of these conditions.(b) Find the eigenvalues and eigenfunctions of the problem.(c) Use a graphing
Determine whether it is possible to find values y0 and y1 (Problem 31) and values of L > 0 (Problem 32) so that the given boundary-value problem has (a) Precisely one nontrivial solution,(b) More than one solution,(c) No solution,(d) The trivial solution.y'' + 16y = 0, y(0) = 1, y(L) = 1
Determine whether it is possible to find values y0 and y1 (Problem 31) and values of L > 0 (Problem 32) so that the given boundary-value problem has (a) Precisely one nontrivial solution,(b) More than one solution,(c) No solution,(d) The trivial solution.y'' + 16y = 0, y(0) = y0, y(π/2) =
Assume that the model for the spring/mass system in Problem 29 is replaced bym'' + 2x' + kx = 0In other words, the system is free but is subjected to damping numerically equal to 2 times the instantaneous velocity. With the same initial conditions and spring constant as in Problem 29, investigate
The temperature u(r) in the circular ring shown in Figure 5.2.11 is determined from the boundary-value problemFIGURE 5.2.11where u0 and u1 are constants. Show that d'u du dr 0, u(a) = u, u(b) = u1, dr
Consider two concentric spheres of radius r = a and r = b, a < b. See Figure 5.2.10. The temperature u(r) in the region between the spheres is determined from the boundaryvalue problemwhere u0 and u1 are constants. Solve for u(r).Figure 5.2.10. d'u du + 2 0, , и(а) %3D ио. и(b) %3D Uo, u(b)
When the magnitude of tension T is not constant, then a model for the deflection curve or shape y(x) assumed by a rotating string is given bySuppose that 1 < x < e and that T(x) = x2.(a) If y(1) = 0, y(e) = 0, and ρω2 > 0.25, show that the critical speeds of angular rotation areωn = ½
Consider the boundary-value problem introduced in the construction of the mathematical model for the shape of a rotating string:For constant T and ρ, define the critical speeds of angular rotation ωn as the values of v for which the boundary value problem has nontrivial solutions. Find the
Suppose that a uniform thin elastic column is hinged at the end x = 0 and embedded at the end x = L.(a) Use the fourth-order differential equation given in Problem 23 to find the eigenvalues ln, the critical loads Pn, the Euler load P1, and the deflections yn(x).(b) Use a graphing utility to graph
As was mentioned in Problem 22, the differential equation (5) that governs the deflection y(x) of a thin elastic column subject to a constant compressive axial force P is valid only when the ends of the column are hinged. In general, the differential equation governing the deflection of the column
The critical loads of thin columns depend on the end conditions of the column. The value of the Euler load P1 in Example 4 was derived under the assumption that the column was hinged at both ends. Suppose that a thin vertical homogeneous column is embedded at its base (x = 0) and free at its top (x
Consider Figure 5.2.6. Where should physical restraints be placed on the column if we want the critical load to be P4? Sketch the deflection curve corresponding to this load.Figure 5.2.6. y L. L- (a) (b) (c)
Find the eigenvalues and eigenfunctions for the given boundary-value problem. Consider only the case λ = α4, α > 0.y(4) – λy = 0, y'(0) = 0, y'''(0) = 0, y(π) = 0, y''(π) = 0
Find the eigenvalues and eigenfunctions for the given boundary-value problem. Consider only the case λ = α4, α > 0.y(4) – λy = 0, y(0) = 0, y''(0) = 0, y(1) = 0, y''(1) = 0
Find the eigenvalues and eigenfunctions for the given boundary-value problem.x2y" + xy' + λy = 0, y'(e-1) = 0, y(I) = 0
Find the eigenvalues and eigenfunctions for the given boundary-value problem.x2y" + xy' + λy = 0, y(1) = 0, y(eπ) = 0
Find the eigenvalues and eigenfunctions for the given boundary-value problem.y'' + (λ + 1)y = 0, y'(0) = 0, y'(1) = 0
Find the eigenvalues and eigenfunctions for the given boundary-value problem.y'' + 2y' + (λ + 1)y = 0, y(0) = 0, y(5) = 0
Find the eigenvalues and eigenfunctions for the given boundary-value problem.y" + λy = 0, y(-7) = 0, y(π) = 0
Find the eigenvalues and eigenfunctions for the given boundary-value problem.y'' + λy = 0, y'(0) = 0, y'(π) = 0
Find the eigenvalues and eigenfunctions for the given boundary-value problem.y" + λy = 0, y(0) = 0, y'(π/2) = 0
Find the eigenvalues and eigenfunctions for the given boundary-value problem.y" + λy = 0, y'(0) = 0, y(L) = 0
Find the eigenvalues and eigenfunctions for the given boundary-value problem.y" + λy = 0, y(0) = 0, y(π) = 0
When a compressive instead of a tensile force is applied at the free end of the beam in Problem 7, the differential equation of the deflection isEIy'' = -Py – w(x)x/2.Solve this equation if w(x) = w0 x, 0 < x < L, and y(0) = 0, y'(L) = 0.
A cantilever beam of length L is embedded at its right end, and a horizontal tensile force of P pounds is applied to its free left end. When the origin is taken at its free end, as shown in Figure 5.2.8, the deflection y(x) of the beam can be shown to satisfy the differential equation EIy'' =
(a) Find the maximum deflection of the cantilever beam in Problem 1.(b) How does the maximum deflection of a beam that is half as long compare with the value in part (a)?(c) Find the maximum deflection of the simply supported beam in Problem 2.(d) How does the maximum deflection of the simply

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