dIFFERENTIAL EQUATION

• Solve y’’ + 9y = 0

  A. y = c1 cos3x + c2 sin 3x

  B. y = c1 cos3x - c2 sin 3x

  C. y = e^(3x) (c1 cos3x + c2 sin 3x)

  D. y = e^(3x) (c1 cos3x - c2 sin 3x)
• The differential equation dy/dx = (x^3-2y)/(x) is .

  A. Linear *

  C. Exact

  B. Homogeneous

  D. Separable
• Identify the equation (x+y)dx – (x –y)dy = 0.

  A. Linear

  B. Exact

  C. Homogeneous *

  D. Separable
• Identify the equation dy/dx = (2xy+y^2+1)/(x^2+2xy)

  A. Linear

  B. Exact *

  C. Homogeneous

  D. Separable
• Find the differential equations of the family of lines passing through the origin. A. xdx + ydy = 0

  C. xdy + ydx = 0

  B. xdx – ydy = 0

  D. xdy – ydx = 0 *
• Solve xydy + (1 + y2 )dx = 0.

  A. x 2(1-y2) = C

  B. x 2+y2 = C

  C. x 2(1+y2) = C *

  D. y 2(1+x2) = C
• . Find the differential equation of y = c1e^x + c2e^-x.

  A. y’’ – y = 0 *

  C. y’’ – xy = 0

  B. y’’ – y’ + y = 0

  D. y’’ + y = 0
• Solve (6x – y^2)dx - 2xydy = 0

  A. 3x^2 – xy^2 = C *

  B. 3x^2 + xy^2 = C

  C. x^2 + 2xy^2 = C

  D. x^2 –2 xy^2 = C
• Solve y’’ + 4y’ = 0.

  A. y = c1 e^x + c2 e^-4x

  B. y = c1 + c2 e^-4x

  C. y =c1 x + c2 e^4x

  D. y = c1 x - c2 e^-4x
• Solve xydy + (1 + y^2 )dx = 0.

  A. x^2(1-y^2) = C

  B. x^2+y^2 = C

  C. x^2(1+y^2) = C *

  D. y^2(1+x^2) = C
• A certain radioactive substance has a half-life of 38 hrs. Find how long it takes for 90 % of the radioactivity to be dissipated?

  A. 46 hours

  B. 100 hours

  C. 126 hours *

  D. 152 hours
• A tank initially holds 100 gal of a brine solution containing 20 lb of salt. At t = 0, fresh water is poured into the tank at the rate of 5 gal/min, while the well-stirred mixture leaves the tank at the same rate. Find the amount of salt in the tank at any time t.

  A. 10e^-t/10

  B. 20e^-t/20 *

  C. -10e^t/10

  D. -20e^t/20
• What is the integrating factor to the differential equation dy/dx + y/x = x^2.

  A. x^2

  B. x *

  C. 1/x^2

  D. 1/x
• Solve ty’ + 2y = 4t^2 , given y(1) = 2.

  A. y = 2t^3 + t

  B. y = t + 1/ t

  C. y = t^2 + t + 1

  D. y = t^2 + 1/ t^2 *
• The equation y(x) = c1 sin2x + c2 cos2x is solution to.

  A. y’’ + 4y = 0 *

  B. y’’ + 4xy = sin x

  C. y’’ + y = y’

  D. x y’’ + y = 0
• On a day when the temperature is 30 Celsius, a cool drink is taken from a refrigerator whose temperature is 5C. If the temperature of the drink is 20C after 10 minutes, what will its temperature be after 20 minutes?

  A. 24 degrees

  B. 26 degrees*

  C. 25 degrees

  D. 27 degrees
• Solve (x^2 + y) dx + ( y^3 + x) dy = 0.

  A. (X^3/3)+xy+(y^4/4) = C *

  B. (X^3/3)-2xy+y^4 = C

  C. X^3+2xy+y^4 = C

  D. X^3-2xy+(y^4/4) = C
• Solve dy/dx = (1+y^2)/(1+x^2)

  A. y= (x+C)/(1-Cx)*

  B. y= 2Cx /(1-y)

  C. y= 2x /(1-C)

  D. y= x /(1-C)
• If 20% of a radioactive substance disappears in one year, find its half-life assuming exponential decay.

  A. 5.1 years

  B. 10 years

  C. 3.1 years *

  D. 4 years
• . Solve the equation: dy/dx = 2y/x

  B. y = cx2
• Solve the equation 𝑑𝑦/𝑑𝑥 = (𝑠𝑒𝑐^2𝑦)/(1+𝑥^2)

  
  

  A. 2𝑦 + 𝑠𝑖𝑛 2𝑦 = 4 arctan 𝑥 + C
• Solve the equation: xydx + (x2 + y2 )dy = 0

  D. y^2 (2x^2 + y^2 ) = C
• Find the general equation of the given differential equation: (x+y)dx + (x-y)dy = 0

  B. x^2 + 2xy – y^2 = c
• Find the general equation of the differential equation: 1 + 𝑦^2 + 𝑥��^2 𝑑𝑥 + (𝑥^2𝑦 + 𝑦 + 2𝑥𝑦)𝑑𝑦 = 0

  B. 2𝑥 + 𝑦^2 (1 + 𝑥)^2 = c
• Find the value a that would make the following equation exact: (2ay^2 − 2ye^2x – a) + (2xy − e^2x − 3ay^2 ) y’ = 0.

  B. a = 0.5
• Solve the differential equation: 2(y-4x^2 )dx + xdy = 0

  C. x2y = 2x4 + C
• Solve the differential equation: 𝑑𝑦/𝑑𝑥 − (2/𝑥)𝑦 = (𝑦/𝑥)^3 𝑠𝑖𝑛X

  C. x4 = y^2 (2xcosx – 2sinx + C)
• Solve the equation: dy/dx = 2y/x

  B. y = cx^2
• Solve the equation: xydx + (x^2 + y^2 )dy = 0

  D. y^2 (2x^2 + y^2 ) = C
• What is the order and degree of the following differential equation:

  y(d^2y/dx^2)^4 +2x^3(dy/dx)^5 = 8xy^2

  D. 2,4
• Find the general equation of y” + 3y’ – 10y = 0.

  D. y(t)= C1e^(-5t)+ C2e^(2t)
• Find the general solution of y”- 4y’+4y=0.

  A. 𝑦(t) = 𝐶1𝑒^2𝑡 + 𝐶2𝑡𝑒^2t
• Find the general solution of y” – 4y’ + 9y = 0.

  B. 𝑦(t) = 𝐶1𝑒^2𝑡 cos(sqrt(5𝑡)) + 𝐶2𝑒^2𝑡 sin(sqrt(5𝑡))
• . A radioactive substance has a mass of 100mg. After 10 years it has decayed to a mass of 75 mg. What will be the mass of the substance be after another 10 years?

  A.56.25 mg
• According to Newton’s 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?

  C. 28 F
• . When a bullet is fired into a sand bag, its retardation is assumed equal to the square root of its velocity on entering. For how long will it travel if its velocity on entering the bank is 144 ft /sec?

  A.24 s
• . An object falls from rest in a medium offering a resistance. The velocity of the object before it reaches the ground is given by dV/dt + V/10 = 32. What is the velocity of the object one second after it falls?

  C. 30.45
• Find the differential equation of the family of parabolas having their vertices at the origin and foci on the x axis.

  A. ydx – 2xdy = 0
• A tank contains 200 gallons of salt solution in which 100 lbs of salt are dissolved. A pipe fills the tank with salt solution at the rate of 5 gallons per minute containing 4 lbs of dissolved salt. Assume that the mixture in the tank is kept uniform by stirring. A drain pipe removes the mixture from the tank at the rate of 4 gallon per minute. Determine the amount of salt in the tank after 30 minutes.

  A. 520 lb
• Find the general solution for the Higher Order DE y^iii - y^ii - 2y^i = 0

  c. c1 e^-x +c2 +c3e^2x