Math RefresherUniversity of Santo Tomas In-House Review for ECE

Differential Equations and Advanced Engg Math

1. Determine the solution of the differential equation y + 5y = 0A. y = 5x + CC. y = Ce5xB. y = Ce5x *D. y = 5e5x + C

2. What is the general solution of the differential equation

A. y = sin x + 2 tan x + CB. y = ex 2ex + CC. y = 2x2 x + CD. y = sin 2x + cos 2x + C *

3. In the differential equation with an initial condition x(0) = 12, what is the value of x(2)?A. 3.35 104C. 3.35B. 4.03 103 *D. 6.04

4. A curve passes through the point (1, 1). Determine the absolute value of the slope of the curve at x = 25 if the differential equation of the curve is the exact equation y2dx + 2xydy = 0A. 1/250C. 1/(50sqrt5)B. 1/125D. 1/sqrt(125) *

5. Determine the constant of integration for the separable differential equation xdx + 6y5dy = 0 if x = 0 when y = 2.A. 12C. 24B. 16D. 64 *

6. What is the Laplace transform of e6t?A. 1/(s + 6) *C. e6 + sB. 1/(s 6) D. e6 + s

7. Solve the complex equation (1 + j2)(2 j3) = a + jbA. a = 4, b = 7C. a = 4, b = 7B. a = 4, b = 7D. a = 4, b = 7 *

8. Solve the equation (x j2y) + (y j3x) = 2 + j3A. x = 7, y = 9C. x = 7, y = 9B. x = 7, y = 9 *D. x = 9, y = 7

9. Determine the modulus of .A. 6.785C. 6.875B. 6.786D. 6.986 *

10. The expression (14 + j3)2/5 has how many roots?A. 1C. 3B. 2D. 5 *

11. Determine the principal root of (14 + j3)2/5.A. 0.344967.16 C. 0.3449148.83 B. 0.344976.83D. 0.34494.83 *

12. What is the principal argument of (2 + j)1/4A. 26.45C. 38.36 *B. 36.38D. 45.26

13. If and , solve for the determinant of A BA. 10C. 30B. 20D. 40 *

14. Obtain the value of A. 18C. 32B. 24D. 22 *

15. Determine the value of A. 2 + j30 *C. 2 j30B. 3 j40 D. 3 + j40

16. Which of the following is not a solution of the linear system of equationsx + y + z = 42x 3y + 4z = 333x 2y 2z = 2A. 2C. 5B. 3D. 4 *

17. Obtain the general solution of

A. B. *C. D.

18. Find the curve which satisfies the equation xy = (1 + x2)y and passes through the point (0,1).A. y = (1 + x)1/2C. y = (1 + x2)1/2 *B. y = (1 + x)1/2D. y = (1 + x2)1/2

19. Solve the differential equation y = 2y cos xA. ln y = 2 sin x + c *B. ln y = 2 sin x + cC. ln y = sin x + cD. ln y = sin x + c

20. Solve y y 2y = 0A. y = c1ex + c2e2xC. y = c1ex + c2e2x *B. y = c1ex + c2e2x D. y = c1ex + c2e2x

21. Solve y + 4y + 5y = 0.A. y = e2x(c1 cos x + c2 sin x)B. y = e2x(c1 cos x + c2 sin x) *C. y = ex(c1 cos x + c2 sin x)D. y = ex(c1 cos x + c2 sin x)

22. Solve A. y = c1e2t + c2e2tC. y = c1e2t + c2te2t *B. y = c1e2t + c2e2t D. y = c1e2t + c2te2t

23. Find a solution of the initial-value problem y' = xy3, y(0) = 2.

24. A series RL circuit with R = 12 and L = 4 H is connected to a constant supply of 60 V. The switch is turned on at t = 0. Determine the equation of the current as a function of time. What is the limiting value of the current?A. 3 AC. 5 A *B. 4 AD. 6 A

25. Solve the equation

26. Solve the differential equation A.

C. B.

D.

27. Solve the equation A.

C. *B.

D.

28. Find the orthogonal trajectory of the family of curves x2 y2 = c2

29.

Find the orthogonal trajectories of the family of curves , where k is an arbitrary constant. Ans:

30. A tank contains 20 kg of salt dissolved in 5000 L of water. Brine that contains 0.03 kg of salt per liter of water enters the tank at a rate of 25 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt remains in the tank after half an hour? Ans: 38.1 kg

31. Solve the differential equation

Ans:

32. Find an equation of the curve that passes through the point (0, 1) and whose slope at (x, y) is xy.

33. Find the equation of a curve going through the point (3, 7) having a slope of 4x2 3 at any point in the curve.

34. Find the orthogonal trajectories of the family of curves x2 + 2y2 = k2. Ans: y = Cx2

35. Find the orthogonal trajectories of the family of curves xy = k. Ans: x2 y2 = C

36. A tank contains 1000 L of brine with 15 kg of dissolved salt. Pure water enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank after 20 minutes? Ans: 15e t/100 , 12.3 kg

37. A tank contains 1000 L of pure water. Brine that contains 0.05 kg of salt per liter of water enters the tank at a rate of 5 L/min. Brine that contains 0.04 kg of salt per liter of water enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at a rate of 15 L/min. How much salt is in the tank (a) after minutes and (b) after one hour?

38. A bacteria culture has an initial population of 500. After 4 hours the population has grown to 1000. Assuming the culture grows at a rate proportional to the size of the population, find a function representing the population size after t hours and determine the size of the population after 6 hours. Ans: 1414

39. A radioactive substance has a mass of 100 mg. After 10 years it has decayed to a mass of 75 mg. What will the mass of the substance be after another 10 years? Ans: 56.25 mg

40. How old is a fossil whose 14C/12C ratio is 10 percent of that found in the atmosphere today?Ans: 19,000 years

41. How long will it take a 100-mg sample of 14C to decay to 90 mg? Ans: 871 years

42. An Egyptian papyrus is discovered and it is found that the ratio of 14C to 12C is 65 percent of the known ratio of 14C to 12C in the air today. The half-life of 14C is 5730 years. How old is the papyrus? Ans: 3561 years

43. According to Newtons law of cooling, the temperature of an object changes at a rate proportional to the difference in temperature between the object and the outside medium. If an object whose temperature is 70 Fahrenheit is placed in a medium whose temperature is 20 and is found to be 40 after 3 minutes, what will its temperature be after 6 minutes? Ans: 28F

44. On a day when the temperature is 30 Celsius, a cool drink is taken from a refrigerator whose temperature is 5. If the temperature of the drink is 20 after 10 minutes, what will its temperature be after 20 minutes? Ans: 26C

45. Environmentalists predict that in t years the deer population in a forest will be thousand. What is the population growth rate after three years? Ans: 389 dear/year

46. The rate at which the population of a city grows is given by the differential equation . At what time t does the population reach 900? Ans: 55

47. Solve the differential equation .A.

*C. B.

D. *

48. Find the solution of the initial-value problem A.

*C. B.

D.

49. A series RL circuit with R = 12 and L = 4 H is connected to a supply of E(t) = 60sin30t volts. Find the current after t = 0.5 s. Assume zero initial condition. Ans: 519 mA

50. Find the solution of the differential equation

51. Solve the equation

52. Solve the equation A.

C. *B.

D.

53. Solve the equation A. B. *C.

D.

54. Solve the initial value problem: A. *B.

C.

D.

55. Solve the boundary value problem A.

B.

C. *D.

56. Determine the particular solution of the equation A. x2 + x + C. x2 x *B. x2 x + D. x2 + x

57. Find the Laplace transform of A. 3 + 2x2 B. 5 sin 3x 17e2x C. 2 sin x + 3 cos x D. xe4xE. e2x sin 5x

58. Find the inverse Laplace of the following:A.

B.

C.

59. Solve the differential equation

60. Evaluate (1 + 2i)4

61. Evaluate

62. Evaluate (2 + 3i)i 63. Evaluate ln (1 i)

64. Simplify (4 + 3i)2 + i

65. Given the vectorsA = 3i + 2j k and B = i 3j + 2kDetermine the following:A. A and BB. AB and A BC. angle between A and B

66. Simplify the expression (A B) C, givenA = 3i + 2jB = 2i + 3j + kC = 5i + 2kA. 0C. 20 *B. 15D. 25

67. What is the exponential form of the complex number 3 + 4i?A. 5ei0.6435C. 5ei0.6345B. 5ei0.9273 *D. 5ei0.9237

68. What is the rationalized value of the complex quotient (6 + 2.5i)/(3 + 4i) A. 1.1 0.66iC. 0.32 + 0.66iB. 28 16.5i *D. 0.32 0.66i

69. Given the vector D = (2xyz y2) ax + (x2z 2xy) ay + x2y az, determine D

70. Given the vector field F = xy2z3 ax + x3yz2 ay + x2y3z az, find D

I am not telling you that it would be easy. But I am telling you, it will be worth it!

prepared by Engr. Jefril M. Amboy