Deck 19: Ordinary Differential Equations

Question 1

State the order of the following differential equation and whether it is linear or nonlinear: + cos(x - y) = sin x.

A) 2nd order; linear
B) 2nd order; nonlinear
C) 1st order; nonlinear
D) 1st order; linear
E) None of the above

Answer 1

2nd order; nonlinear

Question 2

State the order of the following differential equation and whether it is linear or nonlinear: + y = sin x

A) 1st order; nonlinear
B) 1st order; linear
C) 2nd order; nonlinear
D) 2nd order; linear
E) None of the above

Answer 2

2nd order; linear

Question 3

State the order of the following differential equation and whether it is linear or nonlinear: - x = 0.

A) 2nd order; linear
B) 2nd order; nonlinear
C) 1st order; nonlinear
D) 1st order; linear
E) None of the above

Answer 3

1st order; nonlinear

Question 4

State the order of the following differential equation and whether it is linear or nonlinear: = 3x.

A) 1st order; linear
B) 1st order; nonlinear
C) 6th order; linear
D) 6th order; nonlinear
E) None of the above

Question 5

State the order of the following differential equation and whether it is linear or nonlinear: = + 2.

A) nth order; linear
B) nth order; nonlinear
C) 1st order; linear
D) 1st order; nonlinear
E) None of the above

Question 6

State the order of the following differential equation and whether it is linear or nonlinear: + y =

A) 2nd order; linear
B) 2nd order; nonlinear
C) 3/2 order; nonlinear
D) 3/2 order; linear
E) None of the above

Question 7

Find two values of r such that y = is a solution of - 9x - 24y = 0, x > 0.

A) r = 6, 4
B) r = -8, 3
C) r = 12, -2
D) r = -12, 2
E) r = -6, -4

Question 8

State the order of the following differential equation and whether it is linear or nonlinear:y = 1.

A) 1st order; linear
B) 1st order; nonlinear
C) 2nd order; nonlinear
D) 2nd order; linear
E) None of the above

Question 9

State the order of the following differential equation and whether it is linear or nonlinear: = xy.

A) 1st order; nonlinear
B) 1st order; linear
C) 2nd order; nonlinear
D) 2nd order; linear
E) None of the above

Question 10

Specify the interval where y = ln x is a solution of the differential equation x + = 0.

A) (- $\infty$ , 0)
B) (-1, 1)
C) (- $\infty$ , $\infty$ )
D) (0, $\infty$ )
E) None of the above

Question 11

Solve the initial-value problem + 4y = 10, y(0) = 200.

A)
B)
C)
D)
E)

Question 12

Solve the differential equation = 6x .

A) x = C
B) x = C
C) x = C
D) x = C
E) x= C

Question 13

Find the solution of the initial-value problem = 3 , y(0) = 0.

A) - 12
B) 9 - 36
C) -
D) - + 12
E) + 12

Question 14

Determine the solution of the differential equation = (1 - 2x) that satisfies the condition

A) y =
B) y =
C) y =
D) y =
E) y =

Question 15

Find the general solution of the differential equation + 3tx = 2t.

A) x = -1 + C
B) x = + C
C) x = 2 + C
D) x = + C
E) x = - + C

Question 16

Find the general solution of the differential equation =

A) x = -2
B) x = 2
C) x = -
D) x =
E) x = + C

Question 17

Find the solution of the initial-value problem = , x(0) = 4.

A) x =
B) x =
C) x =
D) x =
E) x =

Question 18

Find the solution of the initial-value problem = , x(0) = a.

A) x = -ln(1 - - )
B) x = ln(1 + - )
C) x = -ln(1 + - )
D) x = ln(1 - - )
E) x = ln(1 - + )

Question 19

Solve the initial-value problem = - , y(0) = 1.

A) y(x) =
B) y(x) =
C) y(x) =
D) y(x) =
E) y(x) =

Question 20

Find an integrating factor of the form μ(y) for the differential equation and hence find its general solution.

Question 21

Find the function satisfying (x) = x , F(0) = 1.

A) F(x) = -1 - (x + 1)
B) F(x) = 1 - (x - 1)
C) F(x) = 2 - (x + 1)
D) F(x) = 2 + (x - 1)
E) F(x) = 3 - (x + 1)

Question 22

Find the solution of the differential equation = 12 - 2x + 1 that satisfies y(1) = 3.

A) y = - 2
B) 8x - + x + C
C) 8 - + x
D) 8x - + x - 5
E) 8 - + x + 5

Question 23

Solve the initial-value problem x = y (1 + ln(y) - ln(x)), y(1) = .

Question 24

Find the general solution of the differential equation = .

A) y =
B) y =
C) y =
D) y =
E) y =

Question 25

Find the general solution of the differential equation = -9 .

A)
B) -
C) -
D) -
E)

Question 26

The nonexact differential equation M (t, x)) dt + N(t, x) dx = 0 has an integration factor that depends only upon x, if the quantity is independent of t.

Question 27

Given the differential equation (7 + k + 19y) dx + (19x + 2 + 23 )dy = 0, find the constant real numbers k, m, and n so that the differential equation is exact.

A) k = 10, m = -3, n = -15
B) k = -26, m = 3, n = 13
C) k = -6, m = -5, n = 15
D) k = -10, m = -3, n = 15
E) k = 26, m = 3, n = -13

Question 28

Find all solutions to = 1 + .

A) y - (y) = t + C
B) y - (y) = 2t + C
C) y + (y) = t + C
D) y + (y) = 2t + C
E) y + (y) = 2t + C

Question 29

Solve the following initial-value problem: = 9.8 - 0.196v, v(0) = 48.

A) v = 50 -
B) v = 50 - 2
C) v = 50 + 2
D) v = 100 - 4
E) v = 50 +

Question 30

A motorboat is travelling with a velocity of 40 ft/sec when its motor shuts off at time t = 0. Thereafter, its deceleration due to water resistance is given by = -k , where k is a positive constant. After 10 seconds, the boat's velocity is measured to be 20 ft/sec.
(a) What is the value of k?
(b) When will the boat slow to 5 ft/sec?

A) (a) (b) 70 s
B) (a) (b) 700 s
C) (a) (b) 140 s
D) (a) (b) 100 s
E) (a) (b) 50 s

Question 31

The change of dependent variable y = transforms the equation x + y = x cos(x) into

A) + =
B) ln(v) + =
C) + = cos(x)
D) - = cos(x)
E) - = -cos(x)

Question 32

Determine a multiplier (integrating factor) of the first order linear differential equation

A) $\mu$ (x) = -
B) $\mu$ (x) =
C) $\mu$ (x) =
D) $\mu$ (x) = -
E) $\mu$ (x) =

Question 33

Find a general solution of the first order linear differential equation

A) y = - + C
B) y = + C
C) y = - - C
D) y = - C
E) y = + C

Question 34

Find a general solution to the first order linear differential equation

A) y(x) = + + C
B) y(x) = + C
C) y(x) = + C
D) y(x) = + C
E) y(x) = +

Question 35

Find a general solution to the first order linear differential equation

A) y = - ln(1 + ) + C
B) y = ln(1 + ) + C
C) y = ln(1 - ) + C
D) y = - ln(1 - ) + C
E) y = - ln(1 + ) + C

Question 36

Find the general solution of the differential equation

A) y = C +
B) y = C -
C) y = C -
D) y = C +
E) y = C -

Question 37

Solve the initial-value problem

A) x =
B) x =
C) x =
D) x =
E) x =

Question 38

Solve the differential equation = + 2 .

A) y = x ln
B) y = x ln
C) y = x ln
D) y =
E) y =

Question 39

Solve the differential equation (3y + 8x ) dx + (x + 4 y) dy = 0 by finding an integrating factor depending on x.

A) 3xy + 4 = C
B) y + 3 = C
C) y + 2 = C
D) y + 4 = C
E) y + 2 = C

Question 40

A tank contains 200 litres of fluid in which 60 grams of salt is dissolved. A brine solution of 2 grams of salt per litre is pumped into the tank at a rate of 2 litres/minute. The well-stirred solution is then pumped out at the same rate.(a) Write the differential equation in terms of Q(t) and t and write the initial condition to model this problem. (Q(t) should represent the amount of salt in the tank at time t.)(b) Solve the equation in part (a).(c) How much salt will be present in the tank after a long period of time?

A) (a) = 2 - , Q(0) = 60 (b) Q(t) = 200 - 340 (c) 200 grams
B) (a) = 4 - , Q(0) = 60 (b) Q(t) = 400 - 340 (c) 400 grams
C) (a) = 400 - , Q(0) = 60 (b) Q(t) = 40 - 34 (c) 40 grams
D) (a) = 400 - , Q(0) = 60 (b) Q(t) = 4 - 3.4 (c) 4 grams
E) (a) = 200 - , Q(0) = 60 (b) Q(t) = 20 - 34 (c) 20 grams

Question 41

Solve the following initial-value problem:cos(x) + sin(x)y = 2 (x) sin(x) - 1, y( ) = 3 , 0 $\le$ x $\le$ .

A) y = - x - sin(x) + cos(x)
B) y = + x - sin(x) + cos(x)
C) y = - x - sin(x) + 7 cos(x)
D) y = + x - sin(x) + 7 cos(x)
E) y = x - sin(x) + cos(x)

Question 42

Use Euler's method to determine an approximate value of the solution of the initial-value problem using step size

A) 45.6755
B) 43.6755
C) 34.4115
D) 24.5725
E) 64.8978

Question 43

Use Euler's method to determine an approximate value of the solution of the initial-value problem using step size Use formula replication in a computer spreadsheet to carry out the calculations as you now need 100 iterations of the formula.

A) 55.5074
B) 57.7279
C) 60.0371
D) 62.4386
E) 64.8978

Question 44

Use the Euler method to determine an approximate value of the solution of the initial-value problem using step size

A) 110.4464
B) 120.6730
C) 115.2341
D) 112.3558
E) 145.2351

Question 45

Use Euler's method with step size h = 0.1 to approximate the solution of the initial-value problem on the interval

A) 0.5387
B) 0.5633
C) 0.5732
D) 0.5700
E) 0.5123

Question 46

Use the improved Euler method to determine an approximate value of the solution of the initial-value problem using step size

A) 27.3480
B) 40.9407
C) 59.9382
D) 88.7036
E) 64.8978

Question 47

Use the improved Euler method to determine an approximate value of the solution of the initial-value problem using step size Use formula replication in a computer spreadsheet to carry out the calculations as you now need 100 iterations of the formula.

A) 59.8473
B) 62.2892
C) 64.8307
D) 67.2433
E) 64.8978

Question 48

Use the Euler method to determine an approximate value of the solution of the initial-value problem using step size

A) 633.8378
B) 316.6792
C) 156.6864
D) 76.6012
E) 164.3621

Question 49

Use Euler's method with step size h = 0.1 to approximate the solution of the initial-value problem on the interval

A) 0.4662
B) 0.5215
C) 0.5272
D) 0.4978
E) 0.9745

Question 50

Use the Runge-Kutta method to determine an approximate value of the solution of the initial-value problem using step size

A) 144.3001
B) 96.4752
C) 64.8581
D) 43.4739
E) 72.8576

Question 51

Use the Runge-Kutta method to determine an approximate value of the solution of the initial-value problem using step size Use formula replication in a computer spreadsheet to carry out the calculations as you now need 100 iterations of the formula.

A) 59.8473
B) 67.5463
C) 64.8978
D) 70.3027
E) 71.6654

Question 52

Use the Euler method to determine an approximate value of the solution of the initial-value problem using step size

A) 154.9004
B) 151.8943
C) 146.0963
D) 121.3222
E) 138.5736

Question 53

Use the Runge-Kutta method with step size h = 0.1 to approximate the solution of the initial-value problem on the interval

A) 0.5321
B) 0.5380
C) 0.5407
D) 0.5487
E) 0.5582

Question 54

Solve the integral equation y(x) = 1 + dt by reinterpreting it as an initial-value problem.

A) y =
B) y =
C) y =
D) y =
E) y =

Question 55

Solve the initial-value problem + 2 = 0, (2) = 3, y(-1) = -4.

A) y = - + 7x + 12
B) y = - + 7x - 12
C) y = - + x - 2
D) y = - + 7x + 4
E) y = - - x - 4

Question 56

Solve the initial-value problem x + 3 = 0, (1) = 2, y(1) = 3.

A) y = 4 -
B) y = 2 +
C) y = 5 -
D) y = 1 +
E) y = 4 +

Question 57

Solve the initial-value problem = 1 + , (0) = 1, y(0) = 0.

A) y = -ln(cos(x) - sin(x))
B) y = ln(cos(x) - sin(x))
C) y = -ln(sin(x) - cos(x))
D) y = ln(sin(x) - cos(x))
E) None of the above

Question 58

Solve the initial-value problem - 2y = 0, (3) = 1, y(3) = 1.

A) y =
B) y =
C) y =
D) y =
E) y =

Question 59

Given that y = x is a solution of the differential equation , find the solution of the equation that satisfies y(1) = 3 and (1) = 5.

A) y = 3x + 2ln(x)
B) y = x +
C) y = 3x + 2xln(x)
D) y = x +
E) y = 5x -2

Question 60

Verify that y = is a solution of the equation , and find the general solution of this equation

A) y = A + B
B) y = A + B ln x
C) y = A + Bx
D) y = A + B
E) y = A + Bx

Question 61

Find the general solution of the differential equation - + 12y = 0.

A) y = A + B
B) y = A + B
C) y = A + B
D) y = A + B
E) y = A + Bx

Question 62

Find the general solution of the differential equation + 6 + 9y = 0.

A) y = A + B
B) y = (A + Bx)
C) y = (A + Bx)
D) y = Ax + Bx
E) y = A + Bx

Question 63

Find the general solution of the differential equation + + 17y = 0.

A) y = [Acos(4x) + Bsin(4x)]
B) y = [Acos(4x) - Bsin(4x)]
C) y = [Acos(4x) + Bsin(4x)]
D) y = [Acos(4x) - Bsin(4x)]
E) y = [Acos(x) + Bsin(x)]

Question 64

Find the general solution of the differential equation y= 0, where D stands for the differential operator .

Question 65

Find the general solution of the differential equation - 10 + 74y = 0.

A) y = cos(7t) + sin(7t)
B) y = cos(7t) + sin(7t)
C) y = cos(5t) + sin(5t)
D) y = cos(5t) + sin(5t)
E) y = cos(7t) + sin(7t)

Question 66

Find the general solution of the differential equation + 13 + 3y = 0.

A) y = +
B) y = -
C) y = +
D) y = +
E) y = +

Question 67

Solve the initial-value problem 12 - -6y = 0; y(0) = 2, (0) = 10.

A) y(t) = 6 - 8
B) y(t) = 4 - 2
C) y(t) = -6 + 8
D) y(t) = -4 + 6
E) y(t) = -6 + 8

Question 68

Solve the initial-value problem + 2 + 2y = 0; y(0) = 4, (0) = -2.

A) y(t) = 4 cos t - 6 sin t
B) y(t) = 4 cos t - 2 sin t
C) y(t) = 4 cos t + 2 sin t
D) y(t) = 4 cos t + 6 sin t
E) y(t) = 4 cos t + 2 sin t

Question 69

Solve the initial-value problem + 8 + 16y = 0; y(0) = 3, (0) = 1.

A) y = (13t + 3)
B) y = (13t - 3)
C) y = (13t + 3)
D) y = (7t + 1)
E) y = (7t - 1)

Question 70

Solve the initial-value problem + 4 = 0, y(1)= 1, (1) = 2.

A) y = -
B) y = -
C) y = -
D) y = -
E) y = -

Question 71

Find a homogeneous linear second-order differential equation with constant coefficient having general solution y = + t .

A) + 20 - 100y = 0
B) + 20 + 100y = 0
C) - 20 + 100y = 0
D) - 20 - 100y = 0
E) + 10 + 100y = 0

Question 72

Find a homogeneous linear second-order differential equation with constant coefficient having general solution y = .

A) - 2 - y = 0
B) + 2 - y = 0
C) - 2 + y = 0
D) + 2 + 2y = 0
E) + 2 + y = 0

Question 73

Find the general solution of the differential equation - 2y = 0.

A) y = A expt + exp
B) y = A expt + exp
C) y = A exp(t) + exp
D) y = A expt + exp
E) y = A expt + exp

Question 74

Find the general solution of the differential equation - 16y = 0.

A) y = A + B + C cos(2t) + D sin(2t)
B) y = A + B + C cos(2t) + D sin(2t)
C) y = A + B + C cos(4t) + D sin(4t)
D) y = A + B + C cos(2t) + D sin(2t)
E) y = A + B + C cos(4t) + D sin(4t)

Question 75

Find the general solution of the differential equation + 4 + 6 +4 + y = 0.

A) y =
B) y = A + B + Ct + Dt
C) y = A + B + C + D
D) y =
E) y = A + B + C + D

Question 76

Find the general solution of the differential equation + 4y = 0.

A) y = cos(2x) + sin(2x)
B) y = cos(-2x) + sin(-2x)
C) y =
D) y =
E) y =

Question 77

If the auxiliary equation of a certain linear, homogeneous, 7th order DE with constant coefficients factors to find the general solution of the DE.

A) y = A + Bt + C + (D + Et) cos(t) + (F + Gt) sin(t)
B) y = (A + Bt + C ) + (D + Et) cos(t) + (F + Gt) sin(t)
C) y = A + Bt + C + (D + Et) cos(t/2) + (F + Gt) sin(t/2)
D) y = A + Bt + C + (D + Et) cos(t) + (F + Gt) sin(t)
E) y = A + Bt + C + (D + Et) cos(t) + (F + Gt) sin(t)

Question 78

Find the general solution of the differential equation

A) y = Ax + B
B) y = x(A cos(ln x) + B sin(ln x))
C) A + Bx
D) A + B
E) y = x(A + B x )

Question 79

Find the general solution of the differential equation

A) y = Ax cos(ln x) + Bx sin(ln x)
B) y = A cos(2ln x) + B sin(2ln x)
C) y = A cos(ln x) + B sin(ln x)
D) y = Ax + Bx ln x
E) y = Ax cos x + Bx sin x

Question 80

Solve the initial-value problem

A) y = 3t - 2t ln t
B) y = 4t - 3t ln t
C) y = 3t + 2t ln t
D) y = 3t + 2 ln t
E) y = 3t - t ln t