DIFFERENTIAL EQUATIONS
1. Find the order and degree of the differential equation
𝟐
𝒅𝟑 𝒚
𝒅𝒙𝟑
A.
B.
𝟕
−𝟔
𝒅𝟐 𝒚
𝒅𝒙𝟐
𝟑
𝒅𝟒 𝒚
𝟓
𝒅𝒙𝟒
+
+ 𝒚 = 𝐬𝐢𝐧 𝒙
4th order, 7th degree
4th order, 5th degree
2. Find the general solution of
C.
D.
9th order, 13th degree
3rd order, 7th degree
𝒙𝟐 (𝒚 + 𝟏) 𝒅𝒙 + 𝒚𝟐 (𝒙 − 𝟏) 𝒅𝒚 =
𝟎
A.
B.
x2 ‒ y2 + 2x
2ln[(y + 1)/(x
C
x2 + y2 ‒ 2x
2ln[(y + 1)/(x
C
– 2y +
– 1)] =
– 2y +
– 1)] =
3. Find the general solution of
A.
B.
1 ‒ 2x2 = C(1 + y2)
1 + 2x2 = C(1 ‒ y2)
𝒅𝒚
𝒅𝒙
x2 + y2 ‒ 2x + 2y +
2ln[(y + 1)/(x – 1)] = C
C.
x2 + y2 + 2x – 2y +
2ln[(y + 1)/(x – 1)] = C
D.
=
𝟐𝒙
C.
D.
𝒚
𝟐𝒙𝒚𝟐
𝟐𝒙𝟐 𝒚
1 ‒ 2x2 = C(1 ‒ y2)
1 + 2x2 = C(1 + y2)
4. Find the general solution of
A. x3 – 2y3 = Cx
B. 7x3 + 2y3 = Cx
(𝒙𝟑 + 𝒚𝟑 ) 𝒅𝒙 = 𝟑𝒙𝒚𝟐 𝒅𝒚
5. Find the general solution of
A. x(2x2 + 7y2) = C
B. x(2x2 – 7y2) = C
(𝟔𝒙𝟐 − 𝟕𝒚𝟐 ) 𝒅𝒙 − 𝟏𝟒𝒙𝒚 𝒅𝒚 = 𝟎
6. Find the general solution of
C.
D.
C.
D.
‒6x3 – y3 = Cx
‒3x3 + 2y3 = Cx
y(2x2 + 7y2) = C
y(2x2 – 7y2) = C
(𝐜𝐨𝐬 𝒚 + 𝒚𝒄𝒐𝒔 𝒙) 𝒅𝒙 + (𝐬𝐢𝐧 𝒙 −
𝒙 𝐬𝐢𝐧 𝒚) 𝒅𝒚 = 𝟎
A.
B.
x cos y ‒ y sin x = C
‒ x cos y + y sin x = C
7. Find the general solution of
C.
D.
x cos y + y sin x = C
‒ x cos y ‒ y sin x = C
(𝒚𝒆𝒙𝒚 − 𝟐𝒚𝟑 ) 𝒅𝒙 + (𝒙𝒆𝒙𝒚 −
𝟔𝒙𝒚𝟐 − 𝟐𝒚) 𝒅𝒚 = 𝟎
A.
B.
exy + 2xy3 – y2 = C
exy – 2xy3 + y2 = C
C.
D.
exy + 2xy3 + y2 = C
exy – 2xy3 – y2 = C
(𝒙𝟐 + 𝒚𝟐 + 𝟏) 𝒅𝒙 + (𝒙𝟐 −
8. Find the general solution of
𝟐𝒙𝒚) 𝒅𝒚 = 𝟎
A.
B.
x2 ‒ y2 + xy – 1 = Cx
x2 + y2 ‒ xy – 1 = Cx
C.
D.
x2 + y2 + xy – 1 = Cx
x2 ‒ y2 ‒ xy – 1 = Cx
(𝟐𝒙𝒚 − 𝒚𝟐 + 𝒚) 𝒅𝒙 + (𝟑𝒙𝟐 −
9. Find the general solution of
𝟒𝒙𝒚 + 𝟑𝒙) 𝒅𝒚 = 𝟎
A.
B.
x2y3(x – y + 1) = C
xy3(x – y + 1) = C
C.
D.
x3y(x – y + 1) = C
x3y2(x – y + 1) = C
10. Find the general solution of
A. x2y3 ‒ 2 = Cx2y
B. x2y3 + 2 = Cx3y
(𝟐𝒚) 𝒅𝒙 + (𝒙 − 𝒙𝟑 𝒚𝟑 ) 𝒅𝒚 = 𝟎.
11. Find the general solution of
𝒅𝒚
C.
D.
𝒅𝒙
x2y3 + 2 = Cx2y
x2y3 ‒ 2 = Cx3y
+ 𝟐𝒙𝒚 = 𝟒𝒙
A.
y = 2 + Ce
C.
y = 2 + Ce
B.
y = 3 ‒ Ce
D.
y = 2 ‒ Ce
12. Find the general solution of
A. s = te3t (t3/2 + C)
B. s = t2e3t (t2/2 + C)
𝒕 𝒅𝒔 = (𝟑𝒕 + 𝟏)𝒔 𝒅𝒕 + 𝒕𝟑 𝒆𝟑𝒕 𝒅𝒕
13. Find the general solution of
𝒅𝒚
A.
B.
1/(x4y3) = ‒x2 + C
1/(x3y3) = ‒x2 + C
s = t3e3t (t2/2 + C)
s = te3t (t2/2 + C)
𝒚
+ 𝒙 = 𝟑𝒙𝟐 𝒚𝟐
𝒅𝒙
1/(xy) = ‒3(x2/2) + C
1/(xy) = ‒3(x3/2) + C
14. Find the general solution of
A.
B.
C.
D.
C.
D.
𝒅𝒚
1/(xy) = 3(x2/2) + C
1/(xy) = 3(x3/2) + C
𝒚
𝟑 𝒅𝒙 + 𝟑 𝒙 = 𝟐𝒙𝟒 𝒚𝟒
15. Find the general solution of
C.
D.
1/(x3y4) = x2 + C
1/(x4y4) = x2 + C
(𝒙 + 𝒚 + 𝟏) 𝒅𝒙 + (𝒙 − 𝟑𝒚 −
𝟕) 𝒅𝒚 = 𝟎
A.
B.
x2 + 3y2 + 2xy + 2x –
14y = C
x2 – 3y2 ‒ 2xy + 2x –
14y = C
C.
D.
x2 – 3y2 + 2xy + 2x –
14y = C
x2 – 3y2 + 2xy ‒ 2x –
14y = C
Module 10 – DIFFERENTIAL EQUATIONS |
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DIFFERENTIAL EQUATIONS
16. Find the general solution of (𝒙 + 𝟐𝒚 − 𝟏) 𝒅𝒙 + (𝒙 + 𝟐𝒚 + 𝟓) 𝒅𝒚 = 𝟎
x ‒ y + 6 ln(7 – x – 2y)
x + y ‒ 6 ln(7 – x – 2y)
A.
C.
=C
=C
x ‒ y ‒ 6 ln(7 – x – 2y)
x + y + 6 ln(7 – x – 2y)
B.
D.
=C
=C
17. Find the solution of the differential equation (𝑫𝟐
+ 𝟓𝑫 + 𝟔)𝒚 =
𝟎
A.
B.
y = C1e‒3x + C2e‒2x
y = C1e3x + C2e‒2x
C.
D.
y = C1e‒3x + C2e2x
y = C1e3x + C2e2x
18. Find the solution of the differential equation
(𝑫𝟒 − 𝟐𝑫𝟑 +
𝑫𝟐 )𝒚 = 𝟎
A.
B.
y = C1 + C2x + e‒x(C3 +
C4x)
y = C1 + C2x + ex(C3 +
C4x)
C.
D.
y = C1 + C2x + ex(C3 +
C4x2)
y = C1 + C2x2 + e‒x(C3 +
C4x)
19. Find the solution of the differential equation (𝑫𝟐
− 𝟒𝑫 + 𝟓)𝒚 =
𝟎
A.
B.
y = e‒2x(C1 cos x + C2
sin x)
y = e3x(C1 cos x + C2 sin
x)
C.
D.
y = e2x(C1 cos x + C2 sin
x)
y = e‒3x(C1 cos x + C2 sin
x)
20. Find the solution of the differential equation
𝑫𝟓 + 𝟖𝑫𝟑 +
𝟏𝟔𝑫 𝒚 = 𝟎
A.
B.
y = C1 + (C2 + C3x)cos
3x + (C4 + C5x)sin 3x
y = C1 + (C2 + C3x2)cos
2x + (C4 + C5x2)sin 2x
C.
D.
y = C1 + (C2 + C3x2)cos
3x + (C4 + C5x2)sin 3x
y = C1 + (C2 + C3x)cos
2x + (C4 + C5x)sin 2x
21. At every point P(x, y) of a curve, the intercept of the tangent
line on the y-axis is equal to xy2. Find the equation of the curve.
A. x2y + 7x + Cy = 0
C. x2y ‒ 4x + Cy = 0
B. x2y + 5x + Cy = 0
D. x2y – 2x + Cy = 0
22. A man standing in a corner A of a rectangular pool has a boat
in the adjacent corner B on the end of a 6 m long string. He
walks along the side of the pool towards corner C keeping the
string taut. Locate the boat and the man when the former is
3.60 m from AC.
The boat and the man
The boat and the man is
A. is 1.80 m and 6.60 m C. 1.20 m and 6.80 m from
from AB, respectively.
AB, respectively.
The boat and the man
The boat and the man is
B. is 1.60 m and 6.40 m D. 1.40 m and 6.70 m from
from AB, respectively.
AB, respectively.
23. Find the orthogonal trajectories of xy = K.
A. x2 + y2 = C
C. 2x2 + y2 = C
2
2
B. x – y = C
D. x2 – 2y2 = C
24. The population of a certain community follows the law of
exponential change, that is, the rate of change of a certain
quantity being proportional to the quantity at any instant. If the
present population of a community is 144,000 and ten years
ago was 100,000, when will the population double?
A. 25.45 years
C. 19.25 years
B. 32.73 years
D. 18.75 years
25. Radium decomposes at a rate proportional to the amount at
any instant. In 100 years, 100 mg of radium decompose to 96
mg. How many mg will be left after another 100 years?
A. 92.00 mg
C. 91.64 mg
B. 90.23 mg
D. 92.16 mg
26. In Problem 25, determine the “half-life” of radium.
A. 1698 years
C. 1674 years
B. 1689 years
D. 1647 years
27. A boat is being towed at a rate of 16 km/hr. At this instant a
man in the boat rows in the direction of motion with a force of
65 N. If the man and the boat weigh 2000 N and the resistance
to motion is 1.5 times the velocity (m/s) in N, what is the speed
in 1 minute?
A. 19.78 m/s
C. 17.21 m/s
B. 18.35 m/s
D. 16.98 m/s
Module 10 – DIFFERENTIAL EQUATIONS |
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DIFFERENTIAL EQUATIONS
28. A body falls from rest against a resistance proportional to the
square root of the speed at any instant. If the limiting speed
(speed at uniform motion or when the acceleration is zero) is 9
m/s, find the time required to attain a speed of 6.25 m/s.
A. 3.2 secs
C. 1.8 secs
B. 3.9 secs
D. 2.3 secs
33. If the tank in Problem 32 is in the shape of an inverted cone
(diameter across the top = 1.20 m, height = 2.60 m) with the
orifice (diameter = 5 cm) at the vertex. Recompute the time to
empty a full tank.
A. 130 seconds
C. 140 seconds
B. 150 seconds
D. 160 seconds
29. A tank initially contains 200 liters of fresh water. Brine
containing 2.5 N/liter of dissolved salt runs into the tank at the
rate of 8 liters per minute and the mixture kept uniform by
stirring runs out at the rate of 8 liters per minute. How long will
it take for the quantity of salt in the tank to be 180 N?
A. 12.4 minutes
C. 10.6 minutes
B. 13.9 minutes
D. 11.2 minutes
34. Newton’s Law of Cooling states that the rate at which a
substance cools in moving air is proportional to the difference
between the temperature of the substance and that of air. If
the temperature of the air is constant at 30˚F and the
substance cools from 100˚F to 70˚F in 15 minutes, find when
the temperature of the substance will be 40˚F.
A. 67.34 minutes
C. 59.03 minutes
B. 62.45 minutes
D. 52.16 minutes
30. In Problem 29, find the concentration of the salt in the mixture
after 10 minutes.
A. 0.825 N/liter
C. 0.489 N/liter
B. 1.134 N/liter
D. 0.551 N/liter
31. A tank contains 400 liters of brine. Twelve liters of brine, each
liter containing 2.5 N of dissolved salt, enter the tank per
minute, and the mixture, assumed uniform, leaves at the rate
of 8 liters per minute. If the concentration is to be 2 N/liter at
the end of 1 hour, how many Newtons of salt should there be
present in the tank initially?
A. 181 N
C. 175 N
B. 189 N
D. 168 N
35. A constant inductance of 1 H and a variable resistance of 1/(5
+ t) ohms are connected in series with a constant emf of E
volts. If I = 0 at t = 0, what is the value of E if I = 30 A at t =
5 seconds?
A. 8 volts
C. 6 volts
B. 12 volts
D. 24 volts
32. A tank of cross-sectional area A (m2) at water level has an
orifice of area B (m2) at the bottom. If h (m) is the depth of
water in the tank at any time (sec), the rate of flow from the
𝒅𝒉
tank (m3/sec) is given by 𝑨 = −𝟎. 𝟔𝑩 𝟐𝒈𝒉 where g = 9.81
𝒅𝒕
m/s2. Find the time required to empty a cylindrical tank 1.20 m
in diameter and 2.60 m high through a 5 cm diameter orifice
located at the bottom, if the tank is initially full of water.
A. 724 seconds
C. 783 seconds
B. 697 seconds
D. 659 seconds
Module 10 – DIFFERENTIAL EQUATIONS |
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DIFFERENTIAL EQUATIONS
Module 10 – DIFFERENTIAL EQUATIONS |
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DIFFERENTIAL EQUATIONS
Module 10 – DIFFERENTIAL EQUATIONS |
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