MATH 51 Week 6 Not collected, however you are encouraged to approach all problems to prepare for exam A collection of previous exams could be found at the coordinator s web: http://www.math.psu.edu/tseng/class/m51samples.html HW: 3.7: 1,, 7, 11 3.8: 1, 5, 7(a,c) HW: 4.1: 1, 3, 7, 13, 3 4.: 11, 13, 16, 18, 9 1
0 Chapter 3. Second Order Linear Equations Thus we must know the charge on the capacitor and the current in the circuit at some initial time t 0. Alternatively, we can obtain a differential equation for the current I by differentiating Eq. (33) with respect to t, and then substituting for dq/dt from Eq. (31). The result is with the initial conditions From Eq. (3) it follows that LI + RI + 1 C I = E (t), (35) I(t 0 ) = I 0, I (t 0 ) = I 0. (36) I 0 = E(t 0) RI 0 (1/C)Q 0. (37) L Hence I 0 is also determined by the initial charge and current, which are physically measurable quantities. The most important conclusion from this discussion is that the flow of current in the circuit is described by an initial value problem of precisely the same form as the one that describes the motion of a spring mass system. This is a good example of the unifying role of mathematics: Once you know how to solve second order linear equations with constant coefficients, you can interpret the results in terms of mechanical vibrations, electric circuits, or any other physical situation that leads to the same problem. PROBLEMS In each of Problems 1 through 4 determine ω 0, R, and δ so as to write the given expression in the form u = R cos(ω 0 t δ). 1. u = 3 cos t + 4 sin t. u = cos t + 3 sin t 3. u = 4 cos 3t sin 3t 4. u = cos πt 3 sin πt 5. A mass weighing lb stretches a spring 6 in. If the mass is pulled down an additional 3 in. and then released, and if there is no damping, determine the position u of the mass at any time t. Plot u versus t. Find the frequency, period, and amplitude of the motion. 6. A mass of 100 g stretches a spring 5 cm. If the mass is set in motion from its equilibrium position with a downward velocity of 10 cm/s,and if there is no damping,determine the position u of the mass at any time t. When does the mass first return to its equilibrium position? 7. A mass weighing 3 lb stretches a spring 3 in. If the mass is pushed upward, contracting the spring a distance of 1 in., and then set in motion with a downward velocity of ft/s, and if there is no damping, find the position u of the mass at any time t. Determine the frequency, period, amplitude, and phase of the motion. 8. A series circuit has a capacitor of 0.5 10 6 F and an inductor of 1 H. If the initial charge on the capacitor is 10 6 C and there is no initial current, find the charge Q on the capacitor at any time t. 9. A mass of 0 g stretches a spring 5 cm. Suppose that the mass is also attached to a viscous damper with a damping constant of 400 dyn s/cm. If the mass is pulled down an additional cm and then released, find its position u at any time t. Plot u versus t. Determine the quasi frequency and the quasi period. Determine the ratio of the quasi period to the period of the corresponding undamped motion. Also find the time τ such that u(t) < 0.05 cm for all t > τ.
3.7 Mechanical and Electrical Vibrations 03 10. A mass weighing 16 lb stretches a spring 3 in. The mass is attached to a viscous damper with a damping constant of lb s/ft. If the mass is set in motion from its equilibrium position with a downward velocity of 3 in/s, find its position u at any time t. Plot u versus t. Determine when the mass first returns to its equilibrium position. Also find the time τ such that u(t) < 0.01 in for all t > τ. 11. A spring is stretched 10 cm by a force of 3 N. A mass of kg is hung from the spring and is also attached to a viscous damper that exerts a force of 3 N when the velocity of the mass is 5 m/s. If the mass is pulled down 5 cm below its equilibrium position and given an initial downward velocity of 10 cm/s, determine its position u at any time t. Find the quasi frequency µ and the ratio of µ to the natural frequency of the corresponding undamped motion. 1. A series circuit has a capacitor of 10 5 F, a resistor of 3 10, and an inductor of 0. H. The initial charge on the capacitor is 10 6 C and there is no initial current. Find the charge Q on the capacitor at any time t. 13. A certain vibrating system satisfies the equation u + γ u + u = 0. Find the value of the damping coefficient γ for which the quasi period of the damped motion is 50% greater than the period of the corresponding undamped motion. 14. Show that the period of motion of an undamped vibration of a mass hanging from a vertical spring is π L/g, where L is the elongation of the spring due to the mass and g is the acceleration due to gravity. 15. Show that the solution of the initial value problem mu + γ u + ku = 0, u(t 0 ) = u 0, u (t 0 ) = u 0 can be expressed as the sum u = v + w, where v satisfies the initial conditions v(t 0 ) = u 0, v (t 0 ) = 0, w satisfies the initial conditions w(t 0 ) = 0, w (t 0 ) = u 0, and both v and w satisfy the same differential equation as u. This is another instance of superposing solutions of simpler problems to obtain the solution of a more general problem. 16. Show that A cos ω 0 t + B sin ω 0 t can be written in the form r sin(ω 0 t θ). Determine r and θ in terms of A and B. IfR cos(ω 0 t δ) = r sin(ω 0 t θ), determine the relationship among R, r, δ, and θ. 17. A mass weighing 8 lb stretches a spring 1.5 in. The mass is also attached to a damper with coefficient γ. Determine the value of γ for which the system is critically damped; be sure to give the units for γ. 18. If a series circuit has a capacitor of C = 0.8 10 6 F and an inductor of L = 0. H, find the resistance R so that the circuit is critically damped. 19. Assume that the system described by the equation mu + γ u + ku = 0 is either critically damped or overdamped. Show that the mass can pass through the equilibrium position at most once, regardless of the initial conditions. Hint: Determine all possible values of t for which u = 0. 0. Assume that the system described by the equation mu + γ u + ku = 0 is critically damped and that the initial conditions are u(0) = u 0, u (0) = v 0. If v 0 = 0, show that u 0as t but that u is never zero. If u 0 is positive, determine a condition on v 0 that will ensure that the mass passes through its equilibrium position after it is released. 1. Logarithmic Decrement. (a) For the damped oscillation described by Eq. (6), show that the time between successive maxima is T d = π/µ. (b) Show that the ratio of the displacements at two successive maxima is given by exp(γ T d /m). Observe that this ratio does not depend on which pair of maxima is chosen. The natural logarithm of this ratio is called the logarithmic decrement and is denoted by.
3.8 Forced Vibrations 15 Now let us return to Eq. (17) and consider the case of resonance, where ω = ω 0 ; that is,the frequency of the forcing function is the same as the natural frequency of the system. Then the nonhomogeneous term F 0 cos ωt is a solution of the homogeneous equation. In this case the solution of Eq. (17) is u = c 1 cos ω 0 t + c sin ω 0 t + F 0 mω 0 t sin ω 0 t. (4) Because of the term t sin ω 0 t, the solution (4) predicts that the motion will become unbounded as t regardless of the values of c 1 and c ; see Figure 3.8.8 for a typical example. Of course, in reality, unbounded oscillations do not occur. As soon as u becomes large, the mathematical model on which Eq. (17) is based is no longer valid, since the assumption that the spring force depends linearly on the displacement requires that u be small. As we have seen, if damping is included in the model, the predicted motion remains bounded; however, the response to the input function F 0 cos ωt may be quite large if the damping is small and ω is close to ω 0. PROBLEMS In each of Problems 1 through 4 write the given expression as a product of two trigonometric functions of different frequencies. 1. cos 9t cos 7t. sin 7t sin 6t 3. cos πt + cos πt 4. sin 3t + sin 4t 5. A mass weighing 4 lb stretches a spring 1.5 in. The mass is displaced in. in the positive direction from its equilibrium position and released with no initial velocity. Assuming that there is no damping and that the mass is acted on by an external force of cos 3t lb, formulate the initial value problem describing the motion of the mass. 6. A mass of 5 kg stretches a spring 10 cm. The mass is acted on by an external force of 10 sin(t/) N (newtons) and moves in a medium that imparts a viscous force of N when the speed of the mass is 4 cm/s. If the mass is set in motion from its equilibrium position with an initial velocity of 3 cm/s, formulate the initial value problem describing the motion of the mass. 7. (a) Find the solution of Problem 5. (b) Plot the graph of the solution. (c) If the given external force is replaced by a force 4 sin ωt of frequency ω, find the value of ω for which resonance occurs. 8. (a) Find the solution of the initial value problem in Problem 6. (b) Identify the transient and steady state parts of the solution. (c) Plot the graph of the steady state solution. (d) If the given external force is replaced by a force of cos ωt of frequency ω, find the value of ω for which the amplitude of the forced response is maximum. 9. If an undamped spring mass system with a mass that weighs 6 lb and a spring constant 1 lb/in is suddenly set in motion at t = 0 by an external force of 4 cos 7t lb, determine the position of the mass at any time and draw a graph of the displacement versus t. 10. A mass that weighs 8 lb stretches a spring 6 in. The system is acted on by an external force of 8 sin 8t lb. If the mass is pulled down 3 in and then released, determine the position of the mass at any time. Determine the first four times at which the velocity of the mass is zero.
4 Chapter 4. Higher Order Linear Equations equation can be expressed as a linear combination of a fundamental set of solutions y 1,..., y n, it follows that any solution of Eq. () can be written as y = c 1 y 1 (t) + c y (t) + +c n y n (t) + Y(t), (16) where Y is some particular solution of the nonhomogeneous equation (). The linear combination (16) is called the general solution of the nonhomogeneous equation (). Thus the primary problem is to determine a fundamental set of solutions y 1,..., y n of the homogeneous equation (4). If the coefficients are constants, this is a fairly simple problem; it is discussed in the next section. If the coefficients are not constants, it is usually necessary to use numerical methods such as those in Chapter 8 or series methods similar to those in Chapter 5. These tend to become more cumbersome as the order of the equation increases. The method of reduction of order (Section 3.4) also applies to nth order linear equations. If y 1 is one solution of Eq. (4), then the substitution y = v(t)y 1 (t) leads to a linear differential equation of order n 1 for v (see Problem 6 for the case when n = 3). However, if n 3, the reduced equation is itself at least of second order, and only rarely will it be significantly simpler than the original equation. Thus, in practice, reduction of order is seldom useful for equations of higher than second order. PROBLEMS In each of Problems 1 through 6 determine intervals in which solutions are sure to exist. 1. y (4) + 4y + 3y = t. ty + (sin t)y + 3y = cos t 3. t(t 1)y (4) + e t y + 4t y = 0 4. y + ty + t y + t 3 y = ln t 5. (x 1)y (4) + (x + 1)y + (tan x)y = 0 6. (x 4)y (6) + x y + 9y = 0 In each of Problems 7 through 10 determine whether the given set of functions is linearly dependent or linearly independent. If they are linearly dependent, find a linear relation among them. 7. f 1 (t) = t 3, f (t) = t + 1, f 3 (t) = t t 8. f 1 (t) = t 3, f (t) = t + 1, f 3 (t) = 3t + t 9. f 1 (t) = t 3, f (t) = t + 1, f 3 (t) = t t, f 4 (t) = t + t + 1 10. f 1 (t) = t 3, f (t) = t 3 + 1, f 3 (t) = t t, f 4 (t) = t + t + 1 In each of Problems 11 through 16 verify that the given functions are solutions of the differential equation, and determine their Wronskian. 11. y + y = 0; 1, cos t, sin t 1. y (4) + y = 0; 1, t, cos t, sin t 13. y + y y y = 0; e t, e t, e t 14. y (4) + y + y = 0; 1, t, e t, te t 15. xy y = 0; 1, x, x 3 16. x 3 y + x y xy + y = 0; x, x, 1/x 17. Show that W(5, sin t, cos t) = 0 for all t. Can you establish this result without direct evaluation of the Wronskian? 18. Verify that the differential operator defined by L[y] =y (n) + p 1 (t)y (n 1) + +p n (t)y
4.1 General Theory of nth Order Linear Equations 5 is a linear differential operator. That is, show that L[c 1 y 1 + c y ]=c 1 L[y 1 ]+c L[y ], where y 1 and y are n times differentiable functions and c 1 and c are arbitrary constants. Hence, show that if y 1, y,..., y n are solutions of L[y] =0, then the linear combination c 1 y 1 + +c n y n is also a solution of L[y] =0. 19. Let the linear differential operator L be defined by L[y] =a 0 y (n) + a 1 y (n 1) + +a n y, where a 0, a 1,..., a n are real constants. (a) Find L[t n ]. (b) Find L[e rt ]. (c) Determine four solutions of the equation y (4) 5y + 4y = 0. Do you think the four solutions form a fundamental set of solutions? Why? 0. In this problem we show how to generalize Theorem 3..6 (Abel s theorem) to higher order equations. We first outline the procedure for the third order equation y + p 1 (t)y + p (t)y + p 3 (t)y = 0. Let y 1, y, and y 3 be solutions of this equation on an interval I. (a) If W = W(y 1, y, y 3 ), show that y 1 y y 3 W = y 1 y y 3. y 1 y y 3 Hint: The derivative of a 3-by-3 determinant is the sum of three 3-by-3 determinants obtained by differentiating the first, second, and third rows, respectively. (b) Substitute for y 1, y, and y 3 from the differential equation; multiply the first row by p 3, multiply the second row by p, and add these to the last row to obtain W = p 1 (t)w. (c) Show that [ W(y 1, y, y 3 )(t) = c exp ] p 1 (t) dt. It follows that W is either always zero or nowhere zero on I. (d) Generalize this argument to the nth order equation y (n) + p 1 (t)y (n 1) + +p n (t)y = 0 with solutions y 1,..., y n. That is, establish Abel s formula [ ] W(y 1,..., y n )(t) = c exp p 1 (t) dt for this case. In each of Problems 1 through 4 use Abel s formula (Problem 0) to find the Wronskian of a fundamental set of solutions of the given differential equation. 1. y + y y 3y = 0. y (4) + y = 0 3. ty + y y + ty = 0 4. t y (4) + ty + y 4y = 0
4. Homogeneous Equations with Constant Coefficients 31 In determining the roots of the characteristic equation, it may be necessary to compute the cube roots, the fourth roots, or even higher roots of a (possibly complex) number. This can usually be done most conveniently by using Euler s formula e it = cos t + i sin t and the algebraic laws given in Section 3.3. This is illustrated in the following example. EXAMPLE 4 Find the general solution of The characteristic equation is y (4) + y = 0. (0) r 4 + 1 = 0. To solve the equation, we must compute the fourth roots of 1. Now 1, thought of as a complex number, is 1 + 0i. It has magnitude 1 and polar angle π. Thus 1 = cos π + i sin π = e iπ. Moreover, the angle is determined only up to a multiple of π. Thus 1 = cos(π + mπ) + i sin(π + mπ) = e i(π+mπ), where m is zero or any positive or negative integer. Thus ( π ( 1) 1/4 = e i(π/4+mπ/) = cos 4 + mπ ) + i sin ( π 4 + mπ ). The four fourth roots of 1 are obtained by setting m = 0, 1,, and 3; they are 1 + i, 1 + i, 1 i, 1 i. It is easy to verify that, for any other value of m, we obtain one of these four roots. For example, corresponding to m = 4, we obtain (1 + i)/. The general solution of Eq. (0) is ( y = e t/ t c 1 cos + c sin t ) ( + e t/ t c 3 cos + c 4 sin t ). (1) In conclusion, we note that the problem of finding all the roots of a polynomial equation may not be entirely straightforward, even with computer assistance. For instance, it may be difficult to determine whether two roots are equal or merely very close together. Recall that the form of the general solution is different in these two cases. If the constants a 0, a 1,..., a n in Eq. (1) are complex numbers, the solution of Eq. (1) is still of the form (4). In this case, however, the roots of the characteristic equation are, in general, complex numbers, and it is no longer true that the complex conjugate of a root is also a root. The corresponding solutions are complex-valued. PROBLEMS In each of Problems 1 through 6 express the given complex number in the form R(cos θ + i sin θ) = Re iθ. 1. 1 + i. 1 + 3i 3. 3 4. i 5. 3 i 6. 1 i
3 Chapter 4. Higher Order Linear Equations In each of Problems 7 through 10 follow the procedure illustrated in Example 4 to determine the indicated roots of the given complex number. 7. 1 1/3 8. (1 i) 1/ 9. 1 1/4 10. [(cos π/3 + i sin π/3)] 1/ In each of Problems 11 through 8 find the general solution of the given differential equation. 11. y y y + y = 0 1. y 3y + 3y y = 0 13. y 4y y + 4y = 0 14. y (4) 4y + 4y = 0 15. y (6) + y = 0 16. y (4) 5y + 4y = 0 17. y (6) 3y (4) + 3y y = 0 18. y (6) y = 0 19. y (5) 3y (4) + 3y 3y + y = 0 0. y (4) 8y = 0 1. y (8) + 8y (4) + 16y = 0. y (4) + y + y = 0 3. y 5y + 3y + y = 0 4. y + 5y + 6y + y = 0 5. 18y + 1y + 14y + 4y = 0 6. y (4) 7y + 6y + 30y 36y = 0 7. 1y (4) + 31y + 75y + 37y + 5y = 0 8. y (4) + 6y + 17y + y + 14y = 0 In each of Problems 9 through 36 find the solution of the given initial value problem, and plot its graph. How does the solution behave as t? 9. y + y = 0; y(0) = 0, y (0) = 1, y (0) = 30. y (4) + y = 0; y(0) = 0, y (0) = 0, y (0) = 1, y (0) = 0 31. y (4) 4y + 4y = 0; y(1) = 1, y (1) =, y (1) = 0, y (1) = 0 3. y y + y y = 0; y(0) =, y (0) = 1, y (0) = 33. y (4) y 9y + 4y + 4y = 0; y(0) =, y (0) = 0, y (0) =, y (0) = 0 34. 4y + y + 5y = 0; y(0) =, y (0) = 1, y (0) = 1 35. 6y + 5y + y = 0; y(0) =, y (0) =, y (0) = 0 36. y (4) + 6y + 17y + y + 14y = 0; y(0) = 1, y (0) =, y (0) = 0, y (0) = 3 37. Show that the general solution of y (4) y = 0 can be written as y = c 1 cos t + c sin t + c 3 cosh t + c 4 sinh t. Determine the solution satisfying the initial conditions y(0) = 0, y (0) = 0, y (0) = 1, y (0) = 1. Why is it convenient to use the solutions cosh t and sinh t rather than e t and e t? 38. Consider the equation y (4) y = 0. (a) Use Abel s formula [Problem 0(d) of Section 4.1] to find the Wronskian of a fundamental set of solutions of the given equation. (b) Determine the Wronskian of the solutions e t, e t, cos t, and sin t. (c) Determine the Wronskian of the solutions cosh t, sinh t, cos t, and sin t. 39. Consider the spring mass system, shown in Figure 4..4, consisting of two unit masses suspended from springs with spring constants 3 and, respectively. Assume that there is no damping in the system. (a) Show that the displacements u 1 and u of the masses from their respective equilibrium positions satisfy the equations u 1 + 5u 1 = u, u + u = u 1. (i)