DIFFERENTIAL EQUATIONS
CE Review
DIFFERENTIAL EQUATIONS
Exponential Growth and Decay
INSTRUCTION: Select the correct answer for each of the following
questions. Mark only one answer for each item by shading the box
corresponding to the letter of your choice on the answer sheet provided.
STRICTLY NO ERASURES ALLOWED. Use pencil no. 1 only.
MULTIPLE CHOICE
15.
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 milligrams will be left after another 100 years?
a.
88.60 mg
c.
92.16 mg
b.
90.72 mg
d.
95.32 mg
16.
The population of a certain municipality increases at a rate equal 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?
a.
150 years
c.
200 years
b.
180 years
d.
250 years
17.
A certain radioactive element follows the “law of exponential
change” and has a “half-life” of 38 hours. Find (a) how long it takes
for 90% of the radioactivity of the element to be dissipated; (b) the
percentage of radioactivity that remains after 76 hours.
a.
180 hrs, 75 hrs
c.
306 hrs, 75 hrs
b.
128 hrs, 25 hrs
d.
214 hrs, 25 hrs
Order and Degree of a Differential Equation
1.
Determine the order and degree, respectively, of the differential
equation
 d2 y 
 2 
 dx 
a.
b.
3
 dy 

 dx 
+
4
+y=0
1 and 4
2 and 3
c.
d.
4 and 1
3 and 2
Elimination of Arbitrary Constants
2.
3.
Newton’s Law of Cooling
Eliminate the arbitrary constant from the relation cy2 = x2 + y.
a.
2xy dx – (2x2 + y) dy = 0
b.
2xy dx + (2x2 + y) dy = 0
c.
2xy dy – (2x2 + y) dx = 0
d.
2xy dy + (2x2 + y) dx = 0
18.
If a thermometer is taken outdoors where the temperature is 5C
from a room where the temperature is 20C, the reading drops to
10C in one minute. How long after its removal from the room will
the reading be 6C?
a.
2.47 min
c.
4.56 min
b.
3.56 min
d.
5.56 min
19.
At a certain time, a thermometer reading 70 F is taken outdoors
where the temperature is 15 F. Five minutes later, the thermometer
reading is 45 F. After another five minutes, the thermometer is
taken back indoors where the temperature is fixed at 70 F. What is
the thermometer reading ten minutes after it was brought back
indoors? After the thermometer is brought back indoors, when will
the reading be 69 F?
a.
42.93F, 45.22 min
b.
58.50F, 30.14 min
c.
64.37F, 26.89 min
d.
60.43F, 36.72 min
For the equation given below, determine the differential equation by
elimination of arbitrary constants.
y = c1e3x + c2e-x
a.
b.
y” + 2y’ + 3y = 0
y” + 2y’ – 3y = 0
c.
d.
y” – 2y’ – 3y = 0
y” – 2y’ + 3y = 0
Family of Curves
4.
What is the differential equation of the family of lines passing
through the fixed point (h, k)?
a.
(x – h) dx – (y – k) dy = 0
b.
(y – k) dx + (x – h) dy = 0
c.
(x – h) dx + (y – k) dy = 0
d.
(y – k) dx – (x – h) dy = 0
5.
What is the differential equation of the family of lines passing
through the origin?
a.
x dx – y dy = 0
c.
x dx + y dy = 0
b.
y dx + x dy = 0
d.
y dx – x dy = 0
6.
What is the differential equation of the family of parabolas having
their vertices at the origin and their foci on the x-axis?
a.
2x dx – y dy = 0
c.
2x dx + y dy = 0
b.
y dx – 2x dy = 0
d.
y dx + 2x dy = 0
7.
Find the differential equation of the family of circles with center at
(h, k).
a.
(x + h) dx + (y + k) dy = 0
b.
(x + h) dx – (y + k) dy = 0
c.
(x – h) dx + (y – k) dy = 0
d.
(x – h) dx – (y – k) dy = 0
8.
Chemical Solutions
2 2/3
R=
A tank initially contains 200 liters of fresh water. Brine containing
2.50 N/liter of dissolved salt runs into the tank at the rate of 8
liters/min and the mixture kept uniform by stirring runs out at the
same rate. How long will it take for the quantity of salt in the tank to
be 180 N?
a.
11.16 min
c.
13.52 min
b.
17.03 min
d.
15.35 min
•
Unequal Rates of Inflow and Outflow
21.
A tank contains 200 liters of fresh water. Brine containing 2.50
N/liter of dissolved salt runs into the tank at the rate of 8 liters/min
and the mixture kept uniform by stirring runs out at 4 liters/min. Find
the amount of salt when the tank contains 240 liters of brine. The
concentration of salt in the tank after 25 minutes amounts to how
much?
a.
143.67 N, 1.60 N/liter
b.
157.03 N, 1.47 N/liter
c.
183.33 N, 1.39 N/liter
d.
167.43 N, 1.25 N/liter
2 3/2
(1 − y' )
y"
d.
R=
(1 − y' )
y"
Linear Differential Equations with Constant Coefficients
Separation of Variables
10.
Equal Rates of Inflow and Outflow
20.
Find the differential equation of the family of circles with radius R.
(Ans. c)
2 2/3
2 3/2
(1 + y' )
(1 + y' )
R=
R=
a.
c.
y"
y"
b.
9.
•
Solve the equation xydx + (x + 1)dy = 0.
a.
ex = cy(x + 1)
c.
b.
yex = c(x + 1)
d.
y = cex(x + 1)
yex(x + 1) = c
•
Auxiliary Equation with Real, Distinct Roots
22.
A differential equation has an auxiliary equation of the form m3 –
4m2 + m + 6 = 0. What is the general solution of the differential
equation?
a.
y = c1ex + c2e-2x + c3e3x
b.
y = c1e-x + c2e-2x + c3e3x
c.
y = c1ex + c2e-2x + c3e-3x
d.
y = c1e-x + c2e2x + c3e3x
Solve for the particular solution of 2xyy’ = 1 + y2 when x = 2 and y =
3.
a.
1 – y2 = -5x
c.
1 – y2 = 5x
b.
1 + y2 = -5x
d.
1 + y2 = 5x
•
Auxiliary Equation with Real, Repeated Roots
11.
Solve the equation (2x3 – xy2 – 2y + 3)dx – (x2y + 2x)dy = 0.
a.
x4 – 6x – x2y2 – 4xy = c
b.
x4 – 6x + x2y2 – 4xy = c
c.
x4 + 6x + x2y2 – 4xy = c
d.
x4 + 6x – x2y2 – 4xy = c
23.
Find the general solution of (D3 + 3D2 + 3D + 1)y = 0.
a.
y = (c1 + c2x + c3x2)ex
b.
y = (c1 + c2x + c3x2)e-x
c.
y = (c1 + c2x + c3x2)e2x
d.
y = (c1 + c2x + c3x2)e-2x
12.
Solve the equation (cos 2y – 3x2y2) dx + (cos 2y – 2xsin 2y – 2x3y)
dy = 0.
a.
sin 2y + 2x cos 2y – 2x3y2 = c
b.
sin 2y + x cos 2y – x3y2 = c
c.
sin 2y + 2x cos 2y + 2x3y2 = c
d.
sin 2y + x cos 2y + x3y2 = c
24.
Find the general solution of (D4 + 6D3 + 9D2)y = 0.
a.
y = c1 + c2x + c3x2 + c4e3x
b.
y = c1 + c2x + (c3 + c4x)e3x
c.
y = c1 + c2x + (c3 + c4x)e-3x
d.
y = c1 + c2x + c3x2 + c4e-3x
•
Auxiliary Equation with Imaginary Roots
25.
Find the general solution of (D2 + 4)y = 0.
a.
y = c1 cos 2x + c2 sin 2x
b.
y = c1 cos x + c2 sin x
c.
y = e2x(c1 cos x + c2 sin x)
d.
y = ex(c1 cos 2x + c2 sin 2x)
26.
Find the general equation of (D3 – 3D2 + 9D + 13)y = 0.
a.
y = c1e-x + e3x(c2 cos 2x + c3 sin 2x)
b.
y = c1ex + e3x(c2 cos 2x + c3 sin 2x)
c.
y = c1e-x + e2x(c2 cos 3x + c3 sin 3x)
d.
y = c1ex + e2x(c2 cos 3x + c3 sin 3x)
Exact Differential Equations
Linear Equation of Order One
13.
14.
Solve the equation 2(y – 4x2) dx + x dy = 0.
a.
x4 + 2x2y = c
c.
b.
x4 – 2x2y = c
d.
2x4 + x2y = c
2x4 – x2y = c
Solve the equation y dx + (3x – xy + 2) dy = 0.
a.
xy2 = y2 + 4y + 4 + cey
b.
xy3 = y2 + 4y + 4 + cey
c.
xy3 = 2y2 + 4y + 4 + cey
d.
xy2 = 2y2 + 4y + 4 + cey
1
Dindo F. Esplana
DIFFERENTIAL EQUATIONS
CE Review
Applications of Linear Differential Equations
•
Simple Pendulum
27.
Determine the time of oscillation of a pendulum having a length of
12 m long.
a.
0.1581 min
c.
0.1185 min
b.
0.1518 min
d.
0.1158 min
•
Vibration of a Spring
28.
A spring is such that it would be stretched 6 inches by a 12-lb
weight. Let the weight be attached to the spring and pulled down 4
in below the equilibrium point. If the weight is started with an
upward velocity of 2 ft/sec, determine the amplitude of the motion.
a.
4 in
c.
3 in
b.
5 in
d.
6 in
2
Dindo F. Esplana