102
Chapter 6
Differential Equations
Test Form A
Name
__________________________________________
Date
Chapter 6
Class
__________________________________________
Section _______________________
____________________________
1. Find the general solution to the first order differential equation: 2x y 1 yy 0.
1
C
y 12
(a) 2x2 y2 ln y 1 C
(b) x2 y (d) x2 y 12 y2 C
(e) None of these
(c) ln y 1 x2 y C
2. Find the general solution to the first order differential equation: y dx y x dy 0.
(a) y2 C
(b) ln y x C
(d) y x ln x y C
(c) y ln y x Cy
(e) None of these
3. In 1980 the population of a town was 21,000 and in 1990 it was 20,000. Assuming the population
decreases continuously at a rate proportional to the existing population, estimate the population
in the year 2010.
(a) 17,619
(b) 18,000
(c) 19,048
(d) 18,141
(e) None of these
4. A radioactive element has a half-life of 50 days. What percentage of the original sample is
left after 60 days?
(a) 43.53%
(b) 49.56%
(c) 37.50%
(d) 25.00%
(e) None of these
5. Find the particular solution to y sin x given the general solution y C cos x and the
initial condition y
1.
2
(b) 2 cos x
(c) 1 cos x
(d) 1 cos x
(e) None of these
6. Find the orthogonal trajectories for the family of curves y 2 Cx 3.
(a) y 3 x2 K
(d)
3y 2
2x2
(b) 3y 2 2x2 K
(c) 3ky 2x 0
(e) None of these
y
7. The slope field for a differential equation is shown. Choose the equation that
could be a particular solution to that differential equation.
(a) y sin x
(c) y x
(e) y ln x
(b) y 1
x
3
x
−3
3
(d) y e x
−3
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(a) cos x
Chapter 6
Test Bank
103
8. Consider the differential equation y y with the initial condition y 0 2. Use Euler’s method
with h 0.01 to approximate y 0.03.
(a) 2
(b) 1.458
(c) 1.940598
(d) 1.940891
9. Find the general solution to the first-order differential equation
dy
tan xy cos x.
dx
(a) y sec x x C
(e) None of these
(b) 2y y 2 ln sec x sin x C
(c) ln y cos x sin x C
(d) y cos x Cesin x
(e) None of these
dy
y
L
ky 1 . Find the value of b for the logistics
produces y dt
L
1 bekt
dP 3P
P2
differential equation
given the initial condition 0, 15.
dt
20
1200
10. The logistics differential equation
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(a) 15
(b) 80
(c) 12
(d) 11
(e) 79
104
Chapter 6
Differential Equations
Test Form B
Name
__________________________________________
Date
Chapter 6
Class
__________________________________________
Section _______________________
____________________________
1. Find the general solution to the first order differential equation: 4 x dy 2y dx 0.
(a) y C4 x2
(b) 4y x2 y xy C
(d) y4 4 x2 C
(e) None of these
(c) y 4 x2 C
2. Find the general solution to the first order differential equation: xy2 dy x3 y3 dx 0.
(a) 3x 4 8xy3 C
(b) y3 ln Cx
(c) y3 x3 y3 ln x3 C
(d) y3 3x3 ln Cx
(e) None of these
3. A certain type of bacteria increases continuously at a rate proportional to the number present.
If there are 500 present at a given time and 1000 present 2 hours later, how many will there
be 5 hours from the initial time given?
(a) 1750
(b) 2828
(c) 3000
(d) 2143
(e) None of these
y
4. Choose the differential equation that matches the solution curves sketched
in the slope field.
1
x
(b) y ln x
x
4
(c) y x y
(d) y x
(e) y x
−4
5. Find the particular solution to y sin x given the general solution y sin x Ax B and the
initial conditions y
0, y
2.
2
2
(a) sin x 1 (b) sin x 2x (d) sin x 2x 1 (e) None of these
(c) sin x 2x 1 6. Find the solution to the initial value problem e x y xe y with the initial condition y0 0.
2
(a) y ln
ex 1
2
(d) y ln
2
1 ex
2
(b) y ln
2
ex 1
2
2
(e) None of these
(c) y ln
ex 1
2
2
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(a) y 4
Chapter 6
7. Find the general solution of the differential equation: x2 1 tan y
(a) y C
1
dy
x.
dx
(b) y Cx2 1
x2
(d) cos y Cx2 1
Test Bank
(c) cos y C
1
x2
(e) None of these
8. Consider the differential equation y xy with the initial condition y0 1.
Use Euler’s method with h 0.1 to approximate y0.3.
(a) 1
(b) 0.956
(c) 0.198
(d) 0.9702
9. Find the general solution to the first-order differential equation
(b) y x2 ln x x Cx2
(d) 2y y ln x 4 x2 C
(e) None of these
(b) y 2 Cex14
(d) y 2 Cex 12
(e) None of these
2
(c) 6xy x2 C
xy
xy 5.
2
(a) y 2 Cex 14
2
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dy dy
x 1.
dx
x
(a) y Cx2 x 3
10. Solve the Bernoulli equation y (e) None of these
(c) y 1 Cex 12
2
105
106
Chapter 6
Differential Equations
Test Form C
Name
__________________________________________
Date
Chapter 6
Class
__________________________________________
Section _______________________
____________________________
1. Find the particular solution to the differential equation y 3y given the general solution y Ce3x
and the initial condition y 1 20.
(a) y 20e3x3
(b) y 20e3x
(c) y 20ex
(d) y 20e2x
(e) None of these
2. Find the general solution of the differential equation xy 2y 0.
(a) y 2 ln x C
(d) y (b) x2y 2y2 C
C
x2
(c) y 2x C
(e) None of these
3. Find the orthogonal trajectories for the family of curves y x2 C 2 0.
(a) y3 3 ln Kx
(d)
4y2
x2
(b) y2 ln x Ky
K ln x 0
(c) y ln xy2 K
(e) None of these
4. A radioactive element has a half-life of 40 days. What percentage of the original sample is left
after 48 days?
(a) 49.56%
(b) 43.53%
(c) 25.00%
(d) 37.50%
(e) None of these
5. Determine which function is a solution to the differential equation xy 2y 0.
(b)
1
x2
(c) x2
(d) 2 ln
y
x
(e) None of these
6. Determine whether the function f x, y x 3 2x2y 4xy 2 y 3 is homogeneous, and if so,
determine its degree.
(a) Homogeneous; degree 1
(b) Homogeneous; degree 2
(c) Homogeneous; degree 3
(d) Not homogeneous
(e) None of these
dy
y
L
ky 1 . Find the value of b for the logistics
produces y dt
L
1 bekt
dP
7
P2
P
differential equation
given the initial condition 0, 15.
dt
10
300
7. The logistics differential equation
(a) 14
(b) 99
(c) 13
(d) 98
(e) None of these
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(a) e2x
Chapter 6
Test Bank
8. Consider the differential equation y y 2 with the initial condition y1 1. Use Euler’s Method
with h 0.1 to approximate y1.3.
(a) 1.4285714
(b) 1.3700841
(d) 1.221
(e) None of these
(c) 1
9. Find the particular solution of the differential equation y 2y 4, y0 4.
(a) y 4
(b) y 2 2e2x
(d) y 2 e2x
(e) None of these
(c) y 2 2e 2x
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10. Solve the Bernoulli equation y y xy 3.
(a) y 2x 11 Ce (d) y 12
2x
2
2x 1 Cex
(b) y 2x 2Ce 12
(e) None of these
2x
12
(c) y 2x 12 Ce 2x
12
107
108
Chapter 6
Differential Equations
Test Form D
Name
__________________________________________
Date
Chapter 6
Class
__________________________________________
Section _______________________
1. Find the general solution of the differential equation: x cos2 y tan y
____________________________
dy
0.
dx
2. Find the general solution of the differential equation: xy dx x2 y2 dy 0.
3. A certain type of bacteria increases continuously at a rate proportional to the number present.
If there are 500 present at a given time and 1000 present 2 hours later, how many hours
(from the initial given time) will it take for the number of bacteria to be 2500? Round your
answer to 2 decimal places.
4. Sketch a slope field for the differential equation
dy
y
at the points indicated on the graph below.
dx 2x
y
3
2
1
x
−1
1
2
3
−1
5. Verify that the equation y x2 2x 2 Ce x is a solution to the differential equation y y x2 0.
7. Find the solution to the initial value problem: ey cos yy x, y1 0.
8. Consider the differential equation y with h 0.01 to approximate y1.03.
y
1 with the initial condition y1 0. Use Euler’s Method
x
9. Find the general solution to the first-order differential equation xy 2y x2.
10. Find the particular solution of the linear differential equation y y sin x sin x, y
2 2.
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6. Find the orthogonal trajectories of the family y Cx2 and sketch several members of each family.
Chapter 6
Test Bank
109
Test Form E
Name
__________________________________________
Date
Chapter 6
Class
__________________________________________
Section _______________________
____________________________
1. Solve the differential equation: xy y.
y e x
.
x
y
2
2. Find the general solution of the differential equation:
dy
3. Find the particular solution of the differential equation
500 y that satisfies the initial
dx
condition y 0 7.
4. The rate of change of y with respect to x is inversely proportional to the square root of y.
a.
Write a differential equation for the given statement.
b.
Solve the differential equation in part a.
5. The logistics differential equation
dy
y
L
ky 1 produces y . Find the logistics equation that for
dt
L
1 bekt
dP 3P
P2
that satisfies the initial conditions 0, 5.
dt
5
350
6. Find the orthogonal trajectories of the family x2 4y 2 C and sketch several members of each family.
7. Solve the homogenous differential equation: x2 xy y 2 dx x2 dy 0.
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8. Consider the differential equation y 1
with the initial condition y 1 1. Use Euler’s Method
2y
with h 0.1 to approximate y1.3.
9. Solve, by any appropriate method, the first-order differential equation y y tan x tan x.
10. Find the particular solution to the first-order linear differential equation xy y 1 with the
initial condition y 1 15.