Mathematics for Engineers and Scientists (MATH1551)
First Order Ordinary Differential Equations
59. Solve each of the following separable equations:
y
dy
= ,
dx
x
dy
5x + 7
d)
=
,
dx
3y + 2
dy
= y − xy,
g) x
dx
a)
dy
= x3 y 3 ,
dx
dy
3y + 2
e)
=
,
dx
5x + 7
dy
h)
= 1 + x + y + xy.
dx
dy
= 1 + y2,
dx
dy
1 + y2
f)
=
,
dx
1 + x2
b)
c)
60. A hemispherical bowl of radius 2 metres is filled with water which gradually leaks away
through a circular hole of radius 1 millimetre in the bottom of the bowl. Torricelli’s law shows
that if h(t) is the depth in cms of water above the hole after t seconds, then h(t) satisfies the
differential equation
1000h(400 − h)
p
dh
= −6 2gh,
dt
where g = 981.
Calculate the time the bowl takes to empty.
61. (Separable variables) Solve the following equations:
a) (x2 + 1)y
dy
= 1 where y(0) = −3.
dx
dy
= 3x2 e−y where y(−1) = 0.
dx
dy
c)
= sec y where y(0) = 0.
dx
dy
d)
= y 2 sin x where y(π) = −1/5.
dx
b)
62. A wet porous substance in the open air loses its moisture at a rate proportional to the
moisture content. If a sheet hung in the wind loses half its moisture during the first hour, when
will it have lost 99%, assuming weather conditions remaining the same?
63. Solve the following homogeneous equations:
dy
b) 2
=
dx
dy
y y 2
a)
= +
,
dx
x
x
c)
dy
y 2 − x2
=
,
dx
xy
x+y
x
2
,
d)
dy
y 2 − 2xy
= 2
.
dx
x − 2xy
b)
dy
+ y cot x = 2 cos x,
dx
64. Solve the following linear equations:
a)
dy y
+ = x2 ,
dx x
1
c)
dy
+ 2y = ex ,
dx
d) x
dy
= y + x3 .
dx
65. Solve the following linear equations:
a) y 0 − y tan x = sec x where y(0) = 1,
b) xy 0 + y − ex = 0 where y(2) = 1.
66. The circuit below is a condenser consisting of a voltage source V volts, a capacitor C farads
and resistors R and S ohms.
E
C
R
S
The charge Q(t) at time t in the circuit is known to satisfy
RSC
dQ
+ (R + S)Q = SCV.
dt
If R = 5 ohms, S = 10 ohms, C = 10−3 farads and V = 30 volts, calculate Q(t) if Q(0) = 0
coulombs.
67. Write down differential equations governing the current flow in the LR electrical circuits
illustrated below and solve the two equations.
L
a)
L
b)
V (t) = V0
V (t) = V0 sin(ωt)
R
R
Also, find the current at time t in circuit (a) if R = 1 ohm, L = 10 henrys, V0 = 6 volts, and
the current at time t = 0 is 6 amperes.
68. Solve the following Bernoulli equations:
a) y 0 + (y/x) = xy 2 sin x,
b) 2xy 0 = 10x3 y 5 + y,
69. Check that the following O.D.E. is exact and solve it.
x3 + 2y
dy
+ 3x2 y + 1 = 0.
dx
70. Check that the following O.D.E. is exact and solve it.
2y x4 + 1
dy
+ 4x3 y 2 − 2x − 1 = 0.
dx
71. Check that the following O.D.E. is exact and solve it.
x3 cos y
dy
+ 3x2 sin y + 1 = 0.
dx
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c) xy − y 0 = y 3 e−x .
72. Check that the following O.D.E. is exact and solve it.
x2 sec2 y + 1
dy
+ 2x tan y − sin x = 0.
dx
73. Determine if the following differential equations are exact and solve those which are:
a) (1 + x3 )y 0 + 3x2 y + 1 = 0, b) y + (1 + y)y 0 = 0,
d) (5y − 2x)y 0 = x + 2y,
c) (2y + y 0 )e2x = 0,
e) (2y + y 0 )e2x + 1 = 0.
74. Solve the following mixed bag of equations.
a) y 0 = (ex y)2 − y,
c) 2y 0 =
b) y 0 = y 2 − 5y + 6,
y y2
+ ,
x x2
d) y 0 =
1 + sin y
,
cos y
e) x2 y 0 = y(x + y), given that y(1) = −1,
f) y 0 e−x cos x − ye−x sin x = x, given that y(0) = 0,
g) x(x − 1)y 0 + y = x2 e−x .
Second Order Linear O.D.E.s
75. Find the general solution of each of the following.
a) y 00 + 2y 0 − 15y = 0,
b) 2y 00 + 3y 0 − 2y = 0,
c) y 00 − 6y 0 + 25y = 0
d) y 00 + 6y 0 + 9y = 0,
e) y 00 + y 0 − 2y = 2e−x ,
f) y 00 + y 0 − 2y = ex ,
g) y 00 + y 0 − 6y = 52 cos 2x,
h) 2y 00 + 3y 0 − 2y = sin x,
i) y 00 + y = 2 sin x,
j) y 00 − 2y 0 + 2y = ex cos x.
76. Solve 3y 00 + 5y 0 − 2y = 0 with initial conditions y(0) = 4, y 0 (0) = −1.
77. Solve 2y 00 + 5y 0 + 2y = 0 with initial conditions y(0) = 4, y 0 (0) = −1/2.
78. Solve 3y 00 − 88y 0 − 3y = 0 with initial conditions y(0) = 2, y 0 (0) = −4.
79. Solve 3y 00 + 11y 0 − 4y = 0 with initial conditions y(0) = −1, y 0 (0) = −9.
80. Solve y 00 + 2y 0 + 5y = 0 with initial conditions y(0) = 1, y 0 (0) = −3.
81. Solve y 00 − 4y 0 + 29y = 0 with initial conditions y(0) = 2, y 0 (0) = −1.
82. Solve y 00 − 4y 0 13y = 0 with initial conditions y(0) = 1, y 0 (0) = 8.
83. Solve y 00 + 10y 0 + 34y = 0 with initial conditions y(0) = 2, y 0 (0) = −7.
84. Solve y 00 + 2y 0 + y = 0 with initial conditions y(0) = 1, y 0 (0) = 2.
85. Solve y 00 + 6y 0 + 9y = 0 with initial conditions y(0) = 2, y 0 (0) = −5.
86. Solve 4y 00 − 4y 0 + y = 0 with initial conditions y(0) = −3, y 0 (0) = −1/2.
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87. Solve 4y 00 − 12y 0 + 9y = 0 with initial conditions y(0) = 2, y 0 (0) = 6.
88. Solve y 00 − 4y 0 + 5y = 65 cos 2x, with y(0) = 0, y 0 (0) = 0.
89. Solve the following initial value problems.
a) y 00 − 4y 0 + 3y = 0, with y(0) = −1, y 0 (0) = 1,
b) y 00 + 4y 0 + 5y = 0, with y(0) = 1, y 0 (0) = −3,
c) y 00 − 6y 0 + 9y = 0, with y(0) = 2, y 0 (0) = 8,
d) y 00 − y 0 − 2y = 10 sin x, with y(0) = 1, y 0 (0) = 0,
e) y 00 − y 0 − 2y = 3e2x , with y(0) = 0, y 0 (0) = −2.
f) y 00 − 4y 0 + 5y = 65 cos 2x, with y(0) = 0, y 0 (0) = 0.
g) y 00 + y 0 − 6y = −6x − 11, with y(0) = 2, y 0 (0) = 6.
90. Solve y 00 − 2y 0 + 5y = 6 cos x − 2 sin x, with y(0) = 2, y 0 (0) = 4.
91. Solve y 00 − 2y 0 + 5y = 2 sin 3x − 10 cos 3x, with y(0) = 2, y 0 (0) = 2.
92. Solve y 00 − 6y 0 + 10y = 6 sin 2x + 18 cos 2x, with y(0) = 2, y 0 (0) = 3.
93. Solve y 00 − y 0 − 2y = 3e2x , with y(0) = 0, y 0 (0) = −2.
94. Solve y 00 + 2 y 0 + 5 y = 6 + 15 x, with y(0) = 1 , y 0 (0) = −2.
95. Solve y 00 + 2y 0 + y = x2 + 4x + 1, y(0) = 0, y 0 (0) = 1.
96. Solve y 00 + 2y 0 + y = −2e−x , y(0) = 1, y 0 (0) = 1.
97. Let x(t) be the displacement at time t of the mass in a critically damped oscillator with
damping constant c and mass m = 1. If x(0) = 0 and x0 (0) = v0 , show that the mass will come
to rest when x = 2v0 /ce.
98. The differential equation for the current I(t) in an LCR circuit is
L
L
d2 I
dI
I
dV
+
R
+
=
dt2
dt C
dt
V (t)
C
R
Find the steady state and transient currents in the given circuit in the following cases, assuming
that I(0) = I 0 (0) = 0.
a) R=20 ohms, L=10 henrys, C=0.05 farads, V (t) = 50 sin t volts.
b) R=240 ohms, L=40 henrys, C = 10−3 farads, V (t) = 369 sin 10t volts.
c) R=20 ohms, L=5 henrys, C=0.01 farads, V (t) = 850 sin 4t volts.
99. A cylindrical buoy 60 cms in diameter and weight m grams floats in water with its axis
vertical. When depressed slightly and released, it is found that the period of oscillation is 2
seconds. Archimedes’ principle shows that if x(t) is the length of the cylinder immersed at
time t then x(t) satisfies the equation x00 + ax = g, where a = 900πg/m and g = 981. Calculate
the weight of the buoy and the depth of the bottom of the buoy below the waterline when the
buoy is in the equilibrium position.
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