1550078481-Ordinary_Differential_Equations__Roberts_

Answers to Selected Exercises

Chapter 6 APPLICATIONS OF LINEAR

DIFFERENTIAL EQUATIONS WITH

CONSTANT COEFFICIENTS

Exercises 6.1 Second-Order Differential Equations

Exercises 6.1.1 Free Motion

1. P = 4.49 sec F =. 22 cycles/sec

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3. The period of oscillation of both undamped pendula is w = Vilf.,
   which is independent of the mass of the bob. That is, the undamped
   pendula oscillate at the same rate. If both pendula are subject to the
   same damping, then the one with the larger bob oscillates faster.


4. a. y =sin( v'f9.6t + n /2) b. A= 1 P = 1.42 sec
   c. <P = n/2 = 90°
   d. y' = -4.43 rad/sec, y" = O; y' = 4.43 rad/sec, y" = 0


5. a. y = .499e-·^5 t sin( 4.0lt) b.. 041 rad


6. c = .0651 kg/sec 11. m = .8 kg 13. m = 4 kg

15. c < 9.1447616 kg/sec, c = 9.1447616 kg/sec, c > 9.1447616 kg/sec

17. a. simple harmonic motion b. overdamped motion
    c. critically damped motion d. damped oscillatory motion
    e. overdamped oscillatory motion

Exercises 6.1.2 Forced Motion

1. a. w* = 5.715476 cycles/sec
   b. y = .043307 sin(5. 715476t) + .2 cos(5. 715476t) - .247524 sin lOt
   c. FR = .9065468 cycles/sec