A 32-pound weight stretches a spring 6 inches. The weight moves through a medium

The only solution of the initial-value problem y" + ry = 0, y(O) = 0, y' (0) = 0 is __ .

For the method of undetermined coefficients, the assumed form of the particular solution Yp for y" - y = 1 + ex is

A constant multiple of a solution of a linear differential equation is also a solution.

Iff1 andf2 are linearly independent functions on an interval I, then their Wronskian W( f1,f2) * 0 for all x in I.

If a IO-pound weight stretches a spring 2.S feet, a 32-pound weight will stretch it __ feet.

The period of simple harmonic motion of an 8-pound weight attached to a spring whose constant is 6.2S lb/ft is__ seconds.

The differential equation describing the motion of a mass attached to a spring is x' + l6x = 0. If the mass is released at t = 0 from 1 meter above the equilibrium position with a downward velocity of 3 mis, the amplitude of vibrations is meters.

If simple harmonic motion is described by x(t) = (Vi/2) sin ( 2t + cp), the phase angle cp is when x(O) = -! and x'(O) = 1.

Give an interval over whichf1(x) = r andf2(x) = x lxl are linearly independent. Then give an interval on whichf1 and f2 are linearly dependent

Without the aid of the Wronskian determine whether the given set of functions is linearly independent or linearly dependent on the indicated interval. (a) fi(x) = Inx, f2(x) = ln r, (O, oo) (b) fi(x) = X', f2(x) = X'+1 , n = 1, 2, ... , (-oo, oo) (c) fi(x) = x, fi(x) = x + 1, (-oo, oo) (d) fi(x) = cos (x + 7T/2), fi(x) = sin x, (-oo, oo) (e) fi(x) = 0, f2(X) = X, ( -S, S) (f) fi(x) = 2, f2(x) = 2x, (-oo, oo) (g) fi(x) = r, f2(x) = 1 - r, !3(x) = 2 + r, (-oo, 00) (h) fi(x) = xe+1 , f2(x) = ( 4x - S)e, !3(x) = xe, (-oo, 00)

Suppose m1 = 3, m2 = -S, andm3 = 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 if it is (a) an equation with constant coefficients, (b) a Cauchy-Euler equation

FindaCauchy-Eulerdifferentialequationax2y'' + bxy' + cy = 0, where a, b, and c are real constants, if it is known that (a) m1 = 3 and m2 = -1 are roots of its auxiliary equation, (b) m1 = i is a complex root of its auxiliary equation.

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation.y" - 2y' -2y = 0

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation.2y' + 2y' + 3y = 0

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation.y"' + lOy" + 2Sy' = 0

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation.2y" + 9y" + l2y' + Sy = 0

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation.3y" + lOy" + l Sy' + 4y = 0

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation.2y + 3y"' + 2y" + 6y' - 4y = 0

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation.y" - 3y' + Sy = 4x3 - 2x

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation.y" - 2y' + y = re

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation.y"' - Sy" + 6y' = 8 + 2 sin x

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation.y"' - y"=6

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation.y" - 2y' + 2y = e tanx

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation.y - y = ex + e-x

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation.67!-y" + Sxy' - y = 0

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation.2x3 y"' + 19ry" + 39xy' + 9y = o

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation.ry" - 4 xy' + 6y = 2x4 + r

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation.ry" - xy' + Y = x3

Write down theform of the general solution y =Ye+ y P of the given differential equation in the two cases w * a and w = a. Do not determine the coefficients in y P " (a) y" + w2y = sin ax (b) y" - w2y = ea

(a) Given that y = sin x is a solution of y l + 2y"' + l l y" + 2y' + 1 Oy = 0, find the general solution of the DE without the aid of a calculator or a computer. (b) Find a linear second-order differential equation with constant coefficients for which y1 = 1 and y2 = e -x are solutions of the associated homogeneous equation and y P = !r -x is a particular solution of the nonhomogeneous equation.

(a) Write the general solution of the fourth-order DE y 2y" + y = 0 entirely in terms of hyperbolic functions. (b) Write down the form of a particular solution of y

Consider the differential equation x 2 y" - (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 discussed in Section 3.2 leads both to a second solution y2 of the homogeneous equation and to a particular solution y P of the nonhomogeneous equation. Form the general solution of the DE on the interval (0, 00).

In Problems 33-38, solve the given differential equation subject to the indicated conditions.y" - 2y' + 2y = 0, y(7T/2) = 0, y(7T) = -1

In Problems 33-38, solve the given differential equation subject to the indicated conditions.y" + 2y' + y = 0, y( -1) = 0, y'(O) = 0

In Problems 33-38, solve the given differential equation subject to the indicated conditions.y" - y = x + sinx, y(O) = 2, y'(O) = 3

In Problems 33-38, solve the given differential equation subject to the indicated conditions.. y" + y = sec3x, y(O) = 1, y'(O) = !

In Problems 33-38, solve the given differential equation subject to the indicated conditions.y'y" = 4x, y( l) = S, y'(l) = 2

In Problems 33-38, solve the given differential equation subject to the indicated conditions.2y' = 3y2 , y(O) = 1, y'(O) = 1

(a) Use a CAS as an aid in finding the roots of the auxiliary equation for 12y (b) Solve the DE in part (a) subject to the initial conditions y(O) = -1,y'(O) = 2,y"(O) = 5,ym(O) = 0. UseaCASas an aid in solving the resulting systems of four equations in four unknowns.

Find a member of the family of solutions of xy" + y' + Vx = 0 whose graph is tangent to the x-axis atx = 1. Use a graphing utility to obtain the solution curve.

In Problems 41-44, use systematic elimination to solve the given system.dx dy 41. dt + dt = 2x + 2y + 1 42. dx dy dt+2 dt= y+3

In Problems 41-44, use systematic elimination to solve the given system.dx -=2x+y+t- 2 dt dy -= 3x + 4y - 4t dt

In Problems 41-44, use systematic elimination to solve the given system.(D - 2)x -y = - e1 -3x + (D -4)y = -1e'

In Problems 41-44, use systematic elimination to solve the given system.(D + 2).x + (D + l)y = sin2t 5x + (D + 3)y = cos 2t

A free undamped spring/mass system oscillates with a period of 3 s. When 8 lb is removed from the spring, the system then has a period of 2 s. What was the weight of the original mass on the spring?

A 12-pound weight stretches a spring 2 feet. The weight is released from a point 1 foot below the equilibrium position with an upward velocity of 4 ft/s. (a) Find the equation describing the resulting simple harmonic motion. (b) What are the amplitude, period, and frequency of motion? (c) At what times does the weight return to the point 1 foot below the equilibrium position? ( d) At what times does the weight pass through the equilibrium position moving upward? moving downward? (e) What is the velocity of the weight at t = 3?T/16 s? (f) At what times is the velocity zero?

A spring with constant k = 2 is suspended in a liquid that offers a damping force numerically equal to four 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 32-pound weight stretches a spring 6 inches. The weight moves through a medium offering a damping force numerically equal to {J times the instantaneous velocity. Detemrine the values of {J for which the system will exhibit oscillatory motion.

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) Find the equation for the charge at time t. (b) Find the equation for the current at time t. (c) Find the times for which the charge on the capacitor is zero.

Show that the current i(t) in an LRC-series circuit satisfies the differential equation d2i di 1 L dt2 + R dt + C i = E' (t), where E' (t) denotes the derivative of E(t).

Consider the boundary-value problem y" + Ay = 0, y(O) = y(2'1T), y'(O) = y'(2'1T). Show that except forthe case A = 0, there are two independent eigenfunctions corresponding to each eigenvalue

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 w 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 the position of the bead relative to this rotating coordinate system, as shown in FIGURE 3.R.1. In order to apply Newton's second law of motion to this rotating frame of reference it is necessary to use the fact that the net force acting on the bead is the sum of the real forces (in this case, the force due to gravity) and the inertial forces ( coriolis, transverse, and centrifugal). The mathematics is a little complicated, so we give just the resulting differential equation for r, (a) Solve the foregoing DE subject to the initial conditions r(O) = r0, r'(O) = v0 FIGURE 3.R.1 Rotating rod in Problem 52 (b) Determine initial conditions for which the bead exhibits simple harmonic motion. What is the minimum length L of the rod for which it can accommodate simple harmonic motion of the bead? ( c) For initial conditions other than those obtained in part (b ), the bead must eventually fly off the rod. Explain using the solution rl..,t) in part (a). (d) Suppose w = 1 rad/s. Use a graphing utility to plot the graph of the solution rl...t) for the initial conditions rl...0) = 0, r'(O) = v0, where v0 is 0, 10, 15, 16, 16.1, and 17. ( e) Suppose the length of the rod is L = 40 ft. For each pair of initial conditions in part ( d), use a root-finding application to find the total time that the bead stays on the rod.

Suppose a mass m lying on a flat, dry, frictionless surface is attached to the free end of a spring whose constant is k. In FIGURE 3.R.2(a) the mass is shown at the equilibrium position x = 0; that is, the spring is neither stretched nor compressed. As shown in Figure 3.R.2(b), the displacement x(t) of the mass to the right of the equilibrium position is positive and negative to the left. Derive a differential equation for the free horizontal (sliding) motion of the mass. Discuss the difference between the derivation of this DE and the analysis leading to (1) of Section 3.8. rigid support x 0---- (a) Bquilibrium I I I I (b)Motion -x(t) I < 0--+-- 1 x(t) > oRGURE 3.R.2 Sliding spring/mass system in Problem 53

What is the differential equation of motion in Problem 53 if kinetic friction (but no other damping forces) acts on the sliding mass? [Hint: Assume that the magnitude of the force of kinetic friction is A = mg, where mg is the weight of the mass and the constant > 0 is the coefficient of kinetic friction. Then consider two cases: x' > 0 and x' < 0. lnteJpret these cases physically.]

In Problems 55 and 56, use a Green's function to solve the given initial-value problem.y" + y = tanx, y(O) = 2,y'(O) = -5

In Problems 55 and 56, use a Green's function to solve the given initial-value problem.x'lyH - 3xy + 4y = lnx, y(l) = O,y'(l) = 0

Historically, in order to maintain quality control over muniwould tions (bullets) produced by an assembly line, the manufacturer use a ballistic pendulum to determine the muzzle velocity of a gun; that is, the speed of a bullet as it leaves the barrel. The ballistic pendulum, invented in 1742 by the British mathematician and military engineer Benjamin Robins (1707-1751), is simply a plane pendulum consisting of a rod of negligible mass to which a block of wood of mass m,.. is attached. The system is set in motion by the impact of a bullet that is moving horizontally at the unknown muzzle velocity v0; at the time of the impact, t = 0, the combined mass is m,,, + m,,. where m0 is the mass of the bullet embedded in the wood. We have seen in(/) of Section 3.10 that in the case of pendulum small oscillations, the angular displacement 6(t) of a plane shown in Figure 3.11.3 is given by the linear DE 6H + (g/!)6 = 0, where 8 > 0 corresponds to motion to the right of vertical. The velocity v,, can be found by measuring the height h of the mass m,.. + m,, at the maximum displacement angle 9ma. shown in RGURE 3.R.3. Intuitively. the horizontal velocity V of the combined mass mw + m,, after impact is only a fraction of the velocity v,, of the bullet, that is. V = ( m" ) v,.. Now recall, a di.stance m,.. + mb s ttaveled by a particle moving along a circular path is related differentiating to the radius land central angle 8 by the formulas = 18. By the last formula with respect to time t, it follows that the angular velocity "' of the mass and its linear velocity v are related by v = b. Thus the initial angular velocity "'o at the time t at which the bullet impacts the wood block is related to V by V = l"'o or "'o = ( m,, ) vl,,. m.,.,+m" (a) Solve the initial-value problem d2() g dl2 + l() = 0, 8(0) = 0, 8'(0) = "'<>- (b) Use the result from part (a) to show that (c) Use Figure 3.R.3 to express cos9m.u. in terms of land h. Then use the first two terms of the Maclaurin series for cos8 to express Bma. in terms of land h. Finally, show that v,. is given (approximately) by ( m,.. + m") v,, = 4 m,, v2gh. FIGURE 3.R.3 Ballistic pendulum in Problem 57

Use the result in Problem 57 to find the muzzle velocity vb when mb = 5g, m,.. = 1 kg, and h = 6 cm.

The Paris Guns The first mathematically correct theory of projectile motion was originally formulated by Galileo Galilei (1564-1642), then clarified and extended by his younger oo11aborators Bonaventura Cavalieri (1598-1647) and Evangelista Torricelli (1608-1647). Galileo's theory mental was based on two simple hypotheses suggested by experiobservations: that a projectile moves with constant horizontal velocity and with constant downward vertical acceleration. Galileo, Cavalieri, and Torricelli did not have calculus at their disposal, so their arguments were largely geometric, but we can reproduce their results using a system of differential equations. Suppose that a projectile is launched from ground level at an angle 9 with respect to the horizontal and with an initial velocity of magnitude II voll = Vo mis. Let the projectile's height above the ground bey meters and its horizontal distance from the launch site be x meters, and for convenience take the launch site to be the origin in the xy-plane. Then Galileo's hypotheses can be represented by the following initial-value problem: -d2x =O dt2 d2y dt2 = -g, (1) where g = 9.8 m/s2, x(O) = 0, y(O) = 0, x'(O) = v0cos () (the x-component of the initial velocity), and y' (0) = v0 sin () (they-component of the initial velocity). See FIGURE3.R.4 and Problem 23 in Exercises 3.12. y (voCOS 8) l FIGURE 3.R.4 Ballistic projectile (a) Note that the system of equations in (1) is decoupled, that is, it consists of separate differential equations for x(t) and y (t). Moreover, each of these differential equations can be solved simply by anti.differentiating twice. Solve (1) to obtain explicit formulas for x(t) and y(t) in terms of v0 and 8. Then algebraically eliminate t to show that the trajectory of the projectile in the xy-plane is a parabola. (b) A central question throughout the history of ballistics has been this: Given a gun that fires a projectile with a certain initial speed v0, at what angle with respect to the horizontal should the gun be fired to maximize its range? The range is the horizontal distance traversed by a projectile before it hits the ground. Show that according to (1), the range of the projectile is ( v 51 g) sin 2(), so that a maximum range v51g is achieved for the launch angle () = 7T/4 = 45. ( c) Show that the maximum height attained by the projectile iflaunched with() = 45 for maximum.range is v51(4g)

The Paris Guns-Continued Mathematically, Galileo's model in Problem 59 is perlect. But in practice it is only as accurate as the hypotheses upon which it is based. The motion of a real projectile is resisted to some extent by the air, and the stronger this effect, the less realistic are the hypotheses of constant horizontal velocity and constant vertical acceleration, as well as the resulting independence of the projectile's motion in the x and y directions. The first successful model of air resistance was formulated by the Dutch scientist Christiaan Huygens (1629-1695) and Isaac Newton (1643--1727). It was based not so much on a detailed mathematical formulation of the underlying physics involved, which was beyond what anyone could manage at the time, but on physical intuition and groundbreaking experimental work. Newton's version, which is known to this day as Newton's law of air resistance, states that the resisting force or drag force on an object moving through a resisting medium acts in the direction opposite to the direction of the object's motion with a magnitude proportional to the density p of the medium, the cross sectional area A of the object taken perpendicular to the direction of motion, and the square of the speed II vii of the object. In modem vector notation, the drag force is a vector/ v given in terms of the velocity v of the object: where in the coordinate system of Problem 59 the velocity is the vector v = (dxldt, dyldt). When the force of gravity and this drag force are combined according to Newton's second law of motion we get: m (:. ) = (0, - mg) + Iv The initial-value problem (1) is then modified to read: m d2x = _1CpA[ (dx)2 + (dx)2] dx dt2 2 dt dt dt m d2y = -mg - icpA[ (dx)2 + (dx)2] dy dt2 2 dt dt dt' (2) wherex(O) = O,y(O) = O,x'(O) = v0cos8,y'(O) = v0sin8.