06. Basic Engineering Correlation Admath X De 001

Basic Engineering Correlation (Advance Mathematics and Differential Equation Reviewer) 1. Solve the equation y"+6y+9y=0subject to the conditions y(0) =-4 andy (0) = 5. b. y = (-7x-4) e-3x 2. Solve the homogenous equation (x2+y2) dx+2xydy=0 d. x(x2<+3y2) =c 3. The expression equivalent to ∫01+I 6z2 dz is equivalent to d. 4+4i 4. What can be concluded about the function that the graph below depicts?

d. The following shows an odd function. 5. If A = 25 eπ/42 i and B = CiS π\4 then A + B is_____" d. 32.26+23.11i 6. Which of the following power series is a solution to the differential equation y" + y' = 0 ?

a. 7. The differential equation dv = (y2 - 3vy)dy is said to be c. linear in V 8. The laplace transform of t is d. 1/s2 9. Determine the value of the Legendre's polynomial function P2(2). d. P2(2)=1 10. The rate at which a solid substance dissolves varies directly has the amount of undissolved solid present in the solvent and as the difference between the saturation concentration of the substance and the instantaneous concentration of the solution Five grams of A are placed in solvent B .the solution when saturated will hold ten grams of A. If 2 grams of A dossolved in 1 hr, how many grams of A will be in solution in 2 hrs? d. 3 g 11. Find the differential equation whose general solution is y = C1x + C2 ex. d. (x - 1)y" - xy' + y = 0 12. A cylindrical tank is 12ft. In diameter and 8=9 ft high. Water flows into the tank at the rate of /10 cuft/sec. It has a hole radius 1/2 inch at the bottom. The time the tank will be full if initially it is empty is b. 65 min 13. The indicial equation of the Bessel's equation x2 y" + xy' + (x2 - 9) y =0 is a. r² + r - 9 = 0 14. The solution to the equation x2y'+xy'+x2y=0 if x=0.5 is approximately equal to b. 0.5118 15. The different equation y" + 3y' - 4y =2x is c. second order homogenous linear different equation

16. The series a. f(x) = e3x 17. The expression (3+2i)6 is equivalent to a. -2035- 828i

is equivalent to the function

18. What is the order of the differential equation (4 + y")1/3 = e2x d. two 19. The indicial equation of ODE 2xy"+(l+ x)y'-2y=o is a. 2r2 - r =0 20. The population of a certain municipality increases at a rate to the square root of the population. If the present population is 90,000, how long will it take for the population to reach 160,000? c. 200 years 21. Find the equation of the curve at every point at which the tangent line has a slope of 2x. a. y = x2 + C

22. The order of the different equation a. 2 23. A water container whose circular cross section is 6 ft in diameter and whose height is 8 ft. is filled with water. It has a hole at the bottom of radius 1 inch. The time it will take if the tank rests on support so that its 8 ft height is in a horizontal direction and the hole in its bottom is d. 24.95 min 24. Determine the values of the constants r in the indicial equations of the given ordinary differential equation (2x2 — 24"-2,942y = 0 when Frobenius' method is applied. c. = 1,r2= 2

25. Find a power series for the function d. x+x3+x5+...

26. The order of the differential equation is d. 2 27. The expression ∫0πi cos z dz is equivalent to d. 11.55i 28. Solve (cos x cos y - cot x)dx - sin x sin y dy = 0 b. sin x cos y = ln (C sin x) 29. The ganeral solution of the ordinary different equation with c = constant is d. 2 y = 1 + ce-x2 30. The differential equation given is correctly described by which one of the following choices: d2y/dx2 + bxy dy/dx = f(x) d. linear. Second order, non homogenous 31. Which of the following is true about the Fourier coefficients of f(x)= x if -π ≤ x ≤ π the value of f(π/2) is

d. 32. Sugar decomposes in water at a rate proportional to the amount still unchanged. If there were 50 kg of sugar present initially and at the end of 5 hours this is reduced to 20 kg, how long will it take until 90% os the sugar is decomposed. a. 12.56 hr 33. In the higher-order differential equation (4 — x2 )y'''-4y1+y = 0 , x = —2 is a/an point. c. regular 34. Evaluate cos(3 + 5i) c. -73.47 -10.47i 35. A new water pump has a capacity of 60 cu m/day. If its capacity goes down by 15% every year, in how many years will the capacity go down to 20 cu m/day? b. 7.32 yrs.

36. Which of the following is the solution to the Bessel's equation x2 y" + xy' + (x2 - y2) y=0

c. 37. A certain quantity increases at a rate proportional to q itself. If q = 25 when t = 0 and q = 75 when t =2, find q when t = 6. a. 675 38. Calculate the time in hrs, that it will take to reach the fatal conc. Of 40% methane in a kitchen measuring 15 ft x 12.5 ft x8 ft for a leaking stove. The rate of leak is 15 cuft of 100% methane/hr. Assume no fresh air is coming in. The gas rate is measured at the rate conditions prevailing in the kitchen. a. 40 hrs. 39. Determine the Fourier coefficient a() of the function f (x) = 3x2 + 4, —1 < x <1. d. as=10 40. The differential equation (x2 +4xy+y2)dx-xydy=0 is d. homogenous

41. The differential equation can be classified as c. linear but not homogenous 42. A spherical tank whose inner diameter is 2 meters is filled with water (density 1 g/cc). If a tank has a hole 1 cm in diameter at the bottom, the time the tank will be totally empty is d. 6.31 hrs. 43. The simplified form of (3 + 2i) is c. -2,035-828i

44. The radius of conversence of the power series

(not sure yet)

a. 45. Which of the following is a differential equation of the first order of degree one? a. 46. Find the differential equations og the family of lines through the origin. a. xdy - ydx = 0

47. Solve the equation

c. 48. Solve the different equation d. y=(x3 +11)2 49. Determine the general solution of xdy + ydx = 0 c. xy = c 50. Find the equation of the orthogonal trajectories of the system of parabolas y2=2x+C. b. y = C e-x 51. The principal 4th root of 5 + 12i d. 1.82 + 0.55i 52. Evaluate 143 - 41).

d. 1.61- 0.931

53. Solve a. y= -x5+cx6 54. What is the differential equation of a family of parabolas having their vertices at the origin and their vertices on the x-axis? c. 2xdy - ydx = 0 55. When a simple electric circuit, containing no condensers but having inductance and resistance, has the electromotive force removed, the rate of decrease of current is proportional to the current. The current is i amperes t seconds after the cutoff, and i = 40 when t = 0. If the current dies down to 15 amperes in 0.01 sec, fid i after 0.1 sec. d. 0.002amp 56. Solve the differential equation : x(y - x = 1,determine y when x = 2. a. 1.55 57. How can the differential equation a d2x/dt2 + B(t) dx/dt + c = D(t) best be described? d. linear, second order and non homogenous 58. Evaluate sin ( 3 + 4i ) b. 3.85 - 27.02i 59. A body weighing 1960 N is pulled by a constant force of 492 N along a horizontal plane where in the coefficient of friction between the body and the plane id 0.20. Determine the velocity after 20 seconds. d. 9.06 m/s 60. A tank and its contents weigh 100 lbs. The average heat capacity of the system is 0.5 Btu/ lb.F. The liquid in the tank is heated by an immersion heater which delivers 100 Btu/min. Heat is lost from the system at a rate proportional to the difference between the temperature of the system (assumed uniform throughtout at any instant) and the temperature of the surrounding air, the proportionality constant being 2 Btu/minoF. If the air temperature remains constant at 70oF and if the initial temperature of the tank and its contents is 55oF, the temperature of the tank as a function of the is c. T=120-65e-t/25

61. Which of the following is a solution of the wave equation b. u =(x + at)6 62. A low radioactive material is used in biochemical process to induce biological mutation. The isotope is made in the experimental reactor of the Philippine Atomic Energy Commssion, now Philippine Nuclear Research Institute, and ship to the chemical plant. It has a half life of 8.06 days. The plant receive the shipment of the radioactive material which on arrival contain 1 gram of the radioactive material. The plant uses the material at the rate of 0.1 gram per week. The time it will take for the radioactivity to last is b. 3.24 weeks 63. Solve the differential equation dy - xdx = 0, if the curve passes through (1, 0). d. x2 - 2y -1 = 0 64. A 10-ohm resistor and a 5-henry inductor are connected in series with to a 50-volt source at time t = 0. Express the current I as a function of time. c. i = 5(1 - e-2t) 65. Evaluate cosh(5 + 6i) c. 71.25-2073i 66. A 50 lb iron ball is heated to 200oF and then plunged immediately into a vessel containing 100b lbs of water whose temperature is 40oF. The specific heat of iron is 0.11 Btu/lboF. The common temperature, approached by the iron and water as time approaches infinity is c. 48.34oF 67. The rate f decay of radioactivity elements is usually assumed to be proportional to the number of atoms that have not decayed, where λ is the proportionality consatnt. If at time t=0 there are Xo atoms of a given elements, the expression for the number of atoms, X, that have not decayed (as a function of time,t,λ, and Xo) is c. Xoe-λt 68. Which of the following is a term of the power series representation solution of the higher order differential equation 3 y" —2 x y = 0 d. 1

69. The solution to the non homogeneous partial differential equation c. u(x,y)=f(x)e-2y+2y-1 70. Find the general solution of y' = ysec x. d. y = C (sec x + tan x)

71. Determine the value of c such that the function u(x,t) = e -256 sin 2x will be a solution of the heat equation given by c. 8 72. The expression (5+2i)7 is equivalent to c. -116615+60422i 73. Find the principal 5th root of 5+121. d. 1.62+0.391 74. Evaluatecos(2+3i). d. -4.19-9.11i 75. A body whose temperature is 180o is immersed in a liquid which is kept at a constant temperature of 60o. In 10 minutes the temperature of the immersed body decreased to 120o. How long will it take for the body's temperature to decrease to 90o? b. 20 min. 76. the equation y2 = cx is the general solution of c. y' = y/2x

77. Find the radius of the convergence of the series b. |x| < 1/8 78. Radium decomposes at a rate proportional to the amount at any instant. In 100 years, 100 mg of radium decomposes to 96 mg. How many mg will be left after 100 years? c. 92.16 79. A certain subxtance increases at a rate proportional to the square of the instantaneous amount. After 5 days the amount is doubled. Determine the time before the amount is tripled. c. 20/3 80. Evaluate sinh(6 + 5i) a. 57.22 –193.43i 81. Which of the following is true about the Fourier coefficients of

c. ao=10

82. The solution to the homogeneous partial differential equation b. u(x,y)=A(y)e3x +B(y)xe3x 83. Solve xy'(2y -1) = y(1-x) a. ln (xy) = x + 2y + C 84. A tank initialy contains 400 liters of water. Salt solution, containing 1/8 kg of salt per liter of solution flows into the tank at the rate of 8 li/min and the solution, kept well-stirred, flows out of the tank at the rate of 4 li/min. Find the amount of salt in the tank after 100 minutes. c. 75 kg 85. A mothball loses mass by evaporation at rate that is proportional to the surface area. If half tha mass is lost in 100 days, how long will it take the radius to decreases to half its initial value? c. 243 days

86. The laplace transform of et is c. 1/(s-1) 87. Evaluate cosh ( 3 + 5i) a. 2.86 + 9.61i 88. If dy = x2dx, what is the equation of y in terms of x if the curve passes through (1,1)? a. x3 - 3y + 2 = 0 89. Evaluateln(5 +j3). d. 1.76+ j0.54

90. Which of the following power series is a solution to the differential equation

c.

91. Solve the equation b. y =cIe-5x+ c2e-3x Evaluate ln(5 +j3). 1.76+ j0.54 Evaluate cos (2+3i). -4.19-9.11i The coefficient of the x5 term in the Taylor polynomial for sin(2x) is 0.26667 The augmented matrix of the system is given by 111 2 -1 0 520 Which of the following is the solution to the Bessel's equation x² y"" + xy' + (x² - y²) y=0" b.0 specifically b. Using the first three terms of the Taylor polynomial for is equal to 1.321 Which of the following martices is note reduced row echelon form? 1003 0105 0012 For the matrix A = [1 2 0, 4 1 5, 7 5 3] -1 2 75 Find the principal 5th root of 5-2i 1.40 - 0.11i Which of the following matrices is not equivalent to the matrix A = x? 1 -1 0 4 2 4 6 12 5021 Which of the following is not a solution to the equation t3-6-8r=0? 1,82+2.01i If the elements of any two rows of square matrix are exactly the same then the determinant is Unity Find a power series for the function 1+x²+x4+ Using the first three terms of the power series expansion the 0.516