Page 356 - Schaum's Outline of Differential Equations
P. 356
ANSWERS TO SUPPLEMENTARY PROBLEMS 339
boundary conditions 1.49. No values; boundary conditions
1.50. d = -2, c 2 = 3 1.51. d = 0, c 2 = 1
1.52. d = 3, c 2 = -6 1.53. c 1 = 0, c 2 = 1
CHAPTER 2
2.13. Ine volume and the temperature are in direct proportion. As one increases, so will the other; as one decreases,
so will the other.
2.14. The net force acting on a body is proportional the body's acceleration. This assumes the mass is constant.
2.15. Since t is increasing, and T(576) = 0, this model is valid for 576 hours. Any time afterwards gives us a negative
radicand, and hence, an imaginary answer, thereby rendering the model useless.
2.16. At t = 10, because 7"(10) = 0, and T'(t) >0fott> 10.
2.17. The motion must be periodic, because sin 2t is a periodic function of period n.
2.18. (a) 2 cos 2t; (b) -4 sin 2t
2.19. (a) y is a constant; (b) y is increasing; (c) y is decreasing; (d) y is increasing.
3
2.20. — = k(M - X ) , where k is a negative constant.
dt
2.21. The rates of change of gallons of liquid sugar per hour.
2.22. The rates of change of the vats (gal/hr) are affected by the amount of liquid sugar present in the vats, as the equations
reflect. The signs and magnitudes of the constants (a, b, c, d, e, and/) will determine whether there is an increase
or decrease of sugar, depending on the time. The units for a, b, c, and d is (1/hr); the units for e and/is (gal/hr).
CHAPTER 3
x
2
3.15. y' = -y lx 3.16. y' = xl(e - 1)
2
3.17. / = (sin;t-;y -;y) 1/3 3.18. Cannot reduce to standard form
3.19. y' = -y + lnx 3.20. / = 2 and / = x + y + 3
x
3.24. y' = ye- -e x