Questions and Answers of A First Course in Differential Equations

(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
Solve equation (4) subject to the appropriate boundary conditions. The beam is of length L, and w0 is a constant.(a) The beam is simply supported at both ends, and w(x) = w0 x, 0 < x < L.(b)
Solve equation (4) subject to the appropriate boundary conditions. The beam is of length L, and w0 is a constant.(a) The beam is embedded at its left end and simply supported at its right end, and
Solve equation (4) subject to the appropriate boundary conditions. The beam is of length L, and w0 is a constant.(a) The beam is embedded at its left end and simply supported at its right end, and
Solve equation (4) subject to the appropriate boundary conditions. The beam is of length L, and w0 is a constant.(a) The beam is simply supported at both ends, and w(x) = w0, 0 < x < L.(b) Use
Solve equation (4) subject to the appropriate boundary conditions. The beam is of length L, and w0 is a constant.(a) The beam is embedded at its left end and free at its right end, and w(x) = w0, 0
In Problem 57 find the current when the circuit is in resonance.Problem 57Find the charge on the capacitor and the current in an LC-series circuit when E(t) = E0 cos γt V, q(0) = q0 C, and i(0) = i0
Find the charge on the capacitor and the current in an LC-series circuit when E(t) = E0 cos γt V, q(0) = q0 C, and i(0) = i0 A.
Find the charge on the capacitor and the current in an LC-series circuit when L = 0.1 h, C = 0.1 f, E(t) =100 sin γt V, q(0) = 0 C, and i(0) = 0 A.
Show that if L, R, E0, and g are constant, then the amplitude of the steady-state current in Example 10 is a maximum when the capacitance is C = 1/Lγ2.
Show that if L, R, C, and E0 are constant, then the amplitude of the steady-state current in Example 10 is a maximum when γ = 1/√LC. What is the maximum amplitude?
Find the charge on the capacitor in an LRC-series circuit when L = ½ h, R = 10Ω, C = 0.01 f, E(t) = 150 V, q(0) = 1 C, and i(0) = 0 A. What is the charge on the capacitor after a long time?
Find the steady-state current in an LRC-series circuit when L = 1/2h, R = 20 Ω, C = 0.001 f, and E(t) = 100 sin 60t + 200 cos 40t V.
Use Problem 50 to show that the steady-state current in an LRC-series circuit when L = ½ h, R = 20 Ω, C = 0.001 f, and E(t) = 100 sin 60t V, is given by ip(t) = 4.160 sin(60t - 0.588).Problem
Find the steady-state charge and the steady-state current in an LRC-series circuit when L = 1 h, R = 2Ω, C = 0.25 f, and E(t) = 50 cos t V.
Find the charge on the capacitor and the current in the given LRC-series circuit. Find the maximum charge on the capacitor.L = 1 h, R = 100 Ω, C = 0.0004 f, E(t) = 30 V, q(0) = 0 C, i(0) = 2A
Find the charge on the capacitor and the current in the given LRC-series circuit. Find the maximum charge on the capacitor.L = 5/3 h, R = 10 Ω, C = 1/30 f, E(t) = 300 V, q(0) = 0C, i(0) = 0A
Find the charge on the capacitor in an LRC-series circuit when L = ¼ h, R = 20 Ω, C = 1/300 f, E(t) = 0 V, q(0) = 4 C, and i(0) = 0 A. Is the charge on the capacitor ever equal to zero?
Find the charge on the capacitor in an LRC-series circuit at t = 0.01 s when L = 0.05 h, R = 2Ω, C = 0.01 f, E(t) = 0 V, q(0) = 5 C, and i(0) = 0 A. Determine the first time at which the charge on
Consider a driven undamped spring/mass system described by the initial-value problem
(a) Show that the general solution ofIsWhere A = √c21 + c22 and the phase angles ϕ and θ are, respectively, defined by sin ϕ = c1/A, cos ϕ = c2/A andb) The solution in part (a) has the form
Can there be beats when a damping force is added to the model in part (a) of Problem 39? Defend your position with graphs obtained either from the explicit solution of the problemor from solution
(a) Show that x(t) given in part (a) of Problem 39 can be written in the form(b) If we define ε = ½(γ – ω), show that when ε is small an approximate solution isWhen ε is small, the frequency
Compare the result obtained in part (b) of Problem 39 with the solution obtained using variation of parameters when the external force is F0 cos ωt.Problem 39(a) Show that the solution of the
(a) Show that the solution of the initial-value problem(b) Evaluate d?x + w?x = F, cos yt, x(0) = 0, x'(0) = 0 dr? %3D Fo (cos yt - cos wt). w? - y? is x(t)
Solve the given initial-value problem.d2x/dt2 = 9x = 5 sin 3t, x(0) = 2, x'(0) = 0
Solve the given initial-value problem.d2x/dt2 + 4x = -5 sin 2t + 3 cos 2t,x(0) = -1, x'(0) = 1
A mass of 100 grams is attached to a spring whose constant is 1600 dynes/cm. After the mass reaches equilibrium, its support oscillates according to the formula h(t) = sin 8t, where h represents
A mass m is attached to the end of a spring whose constant is k. After the mass reaches equilibrium, its support begins to oscillate vertically about a horizontal line L according to a formula h(t).
In problem 33 write the equation of motion in the form x(t) = Asin(ωt + ϕ) + Be-2tsin(4t + θ). What is the amplitude of vibrations after a very long time?Problem 33When a mass of 2 kilograms is
When a mass of 2 kilograms is attached to a spring whose constant is 32 N/m, it comes to rest in the equilibrium position. Starting at t = 0, a force equal to f(t) = 68e-2t cos 4t is applied to the
A mass of 1 slug, when attached to a spring, stretches it 2 feet and then comes to rest in the equilibrium position. Starting at t = 0, an external force equal to f(t) = 8 sin 4t is applied to the
A mass of 1 slug is attached to a spring whose constant is 5 lb/ft. Initially, the mass is released 1 foot below the equilibrium position with a downward velocity of 5 ft/s, and the subsequent motion
Amass weighing 16 pounds stretches a spring feet. The mass is initially released from rest from a point 2 feet below the equilibrium position, and the subsequent motion takes place in a medium that
A mass weighing 24 pounds stretches a spring 4 feet. The subsequent motion takes place in medium that offers a damping force numerically equal to β(β 0) times the instantaneous velocity. If the
Amass weighing 10 pounds stretches a spring 2 feet. The mass is attached to a dashpot device that offers a damping force numerically equal to β (β > 0) times the instantaneous velocity.
In parts (a) and (b) of Problem 23 determine whether the mass passes through the equilibrium position. In each case find the time at which the mass attains its extreme displacement from the
A1-kilogram mass is attached to a spring whose constant is 16 N/m, and the entire system is then submerged in a liquid that imparts a damping force numerically equal to 10 times the instantaneous
A4-foot spring measures 8 feet long after a mass weighing 8 pounds is attached to it. The medium through which the mass moves offers a damping force numerically equal to √2 times the instantaneous
Amass weighing 4 pounds is attached to a spring whose constant is 2 lb/ft. The medium offers a damping force that is numerically equal to the instantaneous velocity. The mass is initially released
In problem the given figure represents the graph of an equation of motion for a damped spring/mass system. Use the graph to determine(a) Whether the initial displacement is above or below the
In problem the given figure represents the graph of an equation of motion for a damped spring/mass system. Use the graph to determine(a) Whether the initial displacement is above or below the
In problem the given figure represents the graph of an equation of motion for a damped spring/mass system. Use the graph to determine(a) Whether the initial displacement is above or below the
In problem the given figure represents the graph of an equation of motion for a damped spring/mass system. Use the graph to determine(a) Whether the initial displacement is above or below the
A model of a spring/mass system is 4x'' + tx = 0. By inspection of the differential equation only, discuss the behavior of the system over a long period of time.
Amodel of a spring/mass system is 4x'' + e-0.1tx = 0. By inspection of the differential equation only, discuss the behavior of the system over a long period of time.
A certain mass stretches one spring foot and another spring foot. The two springs are attached to a common rigid support in the manner described in Problem 13 and Figure 5.1.16. The first mass is set
Under some circumstances when two parallel springs, with constants k1 and k2, support a single mass, the effective spring constant of the system is given by k = 4k1k2/(k1 + k2). A mass weighing 20
A mass of 1 slug is suspended from a spring whose spring constant is 9 lb/ft. The mass is initially released from a point 1 foot above the equilibrium position with an upward velocity of √3 ft/s.
Amass weighing 64 pounds stretches a spring 0.32 foot. The mass is initially released from a point 8 inches above the equilibrium position with a downward velocity of 5 ft /s.(a) Find the equation of
A mass weighing 8 pounds is attached to a spring. When set in motion, the spring/mass system exhibits simple harmonic motion.(a) Determine the equation of motion if the spring constant is 1 lb/ft and
A mass weighing 32 pounds stretches a spring 2 feet. Determine the amplitude and period of motion if the mass is initially released from a point 1 foot above the equilibrium position with an upward
Another spring whose constant is 20 N/m is suspended from the same rigid support but parallel to the spring/mass system in Problem 6. A mass of 20 kilograms is attached to the second spring, and both
A force of 400 newtons stretches a spring 2 meters. A mass of 50 kilograms is attached to the end of the spring and is initially released from the equilibrium position with an upward velocity of 10
A mass weighing 20 pounds stretches a spring 6 inches. The mass is initially released from rest from a point 6 inches below the equilibrium position.(a) Find the position of the mass at the times t =
Determine the equation of motion if the mass in Problem 3 is initially released from the equilibrium position with a downward velocity of 2 ft /s.Problem 3A mass weighing 24 pounds, attached to the
A mass weighing 24 pounds, attached to the end of a spring, stretches it 4 inches. Initially, the mass is released from rest from a point 3 inches above the equilibrium position. Find the equation of
A 20-kilogram mass is attached to a spring. If the frequency of simple harmonic motion is 2/π cycles/s, what is the spring constant k? What is the frequency of simple harmonic motion if the original
Amass weighing 4 pounds is attached to a spring whose spring constant is 16 lb/ft. What is the period of simple harmonic motion?
Use systematic elimination to solve the given system.(D + 2)x + (D + 1)y = sin 2t5x + (D + 3)y = cos 2t
Use systematic elimination to solve the given system.(D - 2)x - y = -e' -3x + (D - 4) y = -7e'
Use systematic elimination to solve the given system. dx 2x + y + t- 2 dt dy = 3x + 4y – 4t dt
Use systematic elimination to solve the given system. dy 2.x + 2y + 1 dt dx dt dx dy + 2 y + 3 dt dt +
Find a member of the family of solutions of xy'' + y' + √x = 0 whose graph is tangent to the x axis at x = 1. Use a graphing utility to graph the solution curve.
(a) Use a CAS as an aid in finding the roots of the auxiliary equation for12y(4) + 64y''' + 59y'' - 23y' - 12y = 0.Give the general solution of the equation.(b) Solve the DE in part (a) subject to
Solve the given differential equation subject to the indicated conditions.2y'' = 3y2, y(0) = 1, y'(0) = 1
Solve the given differential equation subject to the indicated conditions.y'y'' = 4x, y(1) = 5, y'(1) = 2
Solve the given differential equation subject to the indicated conditions.y'' + y = sec3x, y(0) = 1, y'(0) = 1/2
Solve the given differential equation subject to the indicated conditions.y'' - y = x + sin x, y(0) = 2, y'(0) = 3
Solve the given differential equation subject to the indicated conditions.y'' + 2y' + y = 0, y(-1) = 0, y'(0) = 0
Solve the given differential equation subject to the indicated conditions.y'' - 2y' + 2y = 0, y (π/2) = 0, y(π) = -1
Consider the differential equationx2y'' - (x2 + 2x)y' + (x + 2)y = x3.Verify that y1 = x is one solution of the associated homogeneous equation. Then show that the method of reduction of order
(a) Write the general solution of the fourth-order DE y(4) - 2y'' + y = 0 entirely in terms of hyperbolic functions.(b) Write down the form of a particular solution of y(4) - 2y'' + y = sinh x.
(a) Given that y = sin x is a solution ofy(4) + 2y''' + 11y'' + 2y' + 10y = 0,find the general solution of the DE without the aid of a calculator or a computer.(b) Find a linear second-order
Write down the form of the general solution y = yc + yp of the given differential equation in the two cases ω ≠ α and ω = α. Do not determine the coefficients in yp.(a) y'' + ω2y = sin αx(b)
In problem use the procedures developed in this chapter to find the general solution of each differential equation.x2y'' – xy' + y = x3
In problem use the procedures developed in this chapter to find the general solution of each differential equation.x2y'' - 4xy' + 6y = 2x4 + x2
In problem use the procedures developed in this chapter to find the general solution of each differential equation.2x3y''' + 19x2y'' + 39xy' + 9y = 0
In problem use the procedures developed in this chapter to find the general solution of each differential equation.6x2y'' + 5xy' - y = 0
In problem use the procedures developed in this chapter to find the general solution of each differential equation.y'' – y = 2ex/ex + e-x
In problem use the procedures developed in this chapter to find the general solution of each differential equation.y'' - 2y' + 2y = ex tan x
In problem use the procedures developed in this chapter to find the general solution of each differential equation.y''' – y'' = 6
In problem use the procedures developed in this chapter to find the general solution of each differential equation.y''' - 5y'' + 6y' = 8 + 2 sin x
In problem use the procedures developed in this chapter to find the general solution of each differential equation.y'' - 2y' + y = x2ex
In problem use the procedures developed in this chapter to find the general solution of each differential equation.y'' - 3y' + 5y = 4x3 - 2x
In problem use the procedures developed in this chapter to find the general solution of each differential equation.2y(4) + 3y''' + 2y'' + 6y' - 4y = 0
In problem use the procedures developed in this chapter to find the general solution of each differential equation.3y''' + 10y'' + 15y' + 4y = 0
In problem use the procedures developed in this chapter to find the general solution of each differential equation.2y''' + 9y'' + 12y' + 5y = 0
In problem use the procedures developed in this chapter to find the general solution of each differential equation.y''' + 10y'' + 25y' = 0
In problem use the procedures developed in this chapter to find the general solution of each differential equation.2y'' + 2y' + 3y = 0
In problem use the procedures developed in this chapter to find the general solution of each differential equation.y'' = 2y' - 2y = 0
Consider the differential equation ay'' + by' + cy = g(x), where a, b, and c are constants. Choose the input functions g(x) for which the method of undetermined coefficients is applicable and the
Suppose m1 = 3, m2 = -5, and m3 = 1 are roots of multiplicity one, two, and three, respectively, of an auxiliary equation. Write down the general solution of the corresponding homogeneous linear DE
Without the aid of the Wronskian, determine whether the given set of functions is linearly independent or linearly dependent on the indicated interval.(a) f1(x) = ln x, f2(x) = ln x2, (0, ∞)(b)
Give an interval over which the set of two functions f1(x) = x2 and f2(x) = x |x| is linearly independent. Then give an interval over which the set consisting of f1 and f2 is linearly dependent.
If y1 = ex and y2 = e-x are solutions of homogeneous linear differential equation, then necessarily y = -5e-x + 10ex is also a solution of the DE. _____Answer problem without referring back to the
If yp1 = x is a particular solution of y'' + y = x and yp1 = x2 – 2 is a particular solution of y'' + y = x2, then a particular solution of y'' + y = x2 + x is _____.Answer problem without
yp = Ax2 is particular solution of y''' + y'' = 1 for A = _____.Answer problem without referring back to the text. Fill in the blank or answer true or false.
If y = c1x2 + c2x2 ln x, x > 0, is the general solution of a homogeneous second-order Cauchy Euler equation, then the DE is _______.Answer problem without referring back to the text. Fill in the
If y = 1 – x + 6x2 + 3ex is a solution of a homogeneous fourth-order linear differential equation with constant coefficients, then the roots of the auxiliary equation are _______.Answer problem

Showing 900 - 1000 of 1260