DEPARTMENT OF MATHEMATICAL SCIENCES
FACULTY OF SCIENCE
UNIVERSITI TEKNOLOGI MALAYSIA
SSCE 1793 DIFFERENTIAL EQUATIONS
1.
2.
3.
Solve the following homogeneous equations.
a. 2y 00 + 7y 0 − 4y = 0
b. 4y 00 − 4y 0 + y = 0
c. y 00 + y = 0
d. y 00 − 4y 0 + 7y = 0
Solve the given initial value problems.
a. y 00 + y 0 = 0;
y(0) = 2,
y 0 (0) = 1
b. y 00 − 4y 0 + 3y = 0;
y(0) = 1,
y 0 (0) = 1/3
c. w00 − 4w0 + 2w = 0;
w(0) = 0,
w0 (0) = 1
d. y 00 − 2y 0 + y = 0;
y(0) = 1,
y 0 (0) = −2
e. 2y 00 − 2y 0 + y = 0;
y(0) = −1,
y 0 (0) = 0
Solve the following boundary value problems.
a. y 00 − 10y 0 + 25y = 0;
y(0) = 1,
y(1) = 0
b. y + 4y = 0;
y(0) = 0,
y 0 (π) = 1
c. 2y 00 + y 0 − 6y = 0;
y(0) = 0,
y 0 (1) = 1
00
4.
Solve the following nonhomogeneous equations using the method of undetermined coefficients.
a. 2x0 + x = 3t2 + 10t
b. y00 − y0 + 9y = 3 sin 3x
c. y00 + y0 + y = 2 cos 2x − 3 sin 2x
d. θ00 − 5θ0 + 6θ = rer
e. θ00(t) − θ(t) = t sin t
f. y00 − 2y0 + y = 8ex
g. y00 − y = −11x + 1
h. y00 + 4y = sin 2θ − cos θ
i. y00 + 2y0 + 2y = e
−θ
j. y00 − 3y = x2 − ex
cos θ
k. x00 − 4x0 + 4x = te2t
5.
l. y00 − y = (5x + 1)e3x
Solve the following initial value problems using the method of undetermined coefficients.
a. y 0 − y = 1;
00
y(0) = 1
z(0) = 0,
z 0 (0) = 0
c. y 00 + y 0 − 12y = ex + e2x − 1;
y(0) = 1,
y 0 (0) = 3
d. y 00 − y = sin θ − e2θ ;
y(0) = 1,
y 0 (0) = −1
b. z + z = 2e
6.
TUTORIAL 2
−x
;
Determine the form of a particular solution to the differential equations.
a. y 00 − y = e2x + xe2x + x2 e2x
b. y 00 + 5y 0 + 6y = sin x − cos 2x
c. y 00 − 4y 0 + 5y = e5x + x sin 3x − cos 3x
d. y 00 − 4y 0 + 4y = x2 e2x − e2x
e. y 00 + y 0 − 6yy = e−3x + x cos 3x − sin 2x
2
7.
Solve the following nonhomogeneous equations using the method of variation of parameters.
a. y 00 + y = sec x
b. y 00 + y = cos2 x
c. y 00 − y = cosh x
d. y 00 + y = sec x tan x
1
f. y 00 + 3y 0 + 2y =
1+
ex
ex
00
0
h. y − 2y + y =
1 + x2
j. y 00 + y = 3 sec x − x2 + 1
e. y 00 + 4y = 2 tan x − ex
g. y 00 + 3y 0 + 2y = sin ex
i. y 00 + 2y 0 + y = e−x ln 2x
k. 2y 00 − 2y 0 − 4y = 2e3t
8.
l. y 00 + 4y = csc2 2x
A vibrating string without damping can be modelled by the differential equation
my 00 + ky = 0.
a. If m = 10kg, k = 250kg/sec2 , y(0) = 0.3m and y 0 (0) = −0.1m/sec, find the equation of
motion for this system.
b. When the equation of motion is of the form displayed in (a), the motion is said to
be oscillatory with frequency β/2π. Find the frequency of the oscillation for the spring
system of part (a).
9.
A vibrating string with damping can be modelled by the differential equation
my 00 + by 0 + ky = 0.
a. If m = 10kg, k = 250kg/sec2 , b = 60kg/sec, y(0) = 0.3m and y 0 (0) = −0.1m/sec, find
the equation of motion for this system.
b. Find the frequency of the oscillation.
c. Compare the results of this problem to Question 8 and determine what effect the damping
has on the frequency of oscillation. What other effects does it have on the solution?
10.
The motion of a certain mass-spring system with damping is governed by
y00(t) + 6y0(t) + 16y(t) = 0
y(0) = 1,
y0(0) = 0.
Find the equation of motion.
11.
Determine the equation of motion for an undamped system at resonance governed by
d2 y
+ 9y = 2 cos 3t
dt2
y(0) = 1, y0(0) = 0.
Sketch the solution.
12.
An undamped system is governed by
m
where γ 6= ω =
13.
q
k
m.
d2 y
+ ky = F0 cos γt;
dt2
y(0) = y0(0) = 0.
Find the equation of motion of the system.
Consider the vibrations of a mass-spring system when a periodic force is applied. The system
is governed by the differential equation
mx00 + bx0 + kx = F0 cos γt
where F0 and γ are nonnegative constants, and 0 < b2 < 4mk.
3
a. Show that the general solution to the corresponding homogeneous equation is
!
√
4mk − b2
(−b/m)t
xh (t) = Ae
sin
t+φ .
2m
b. Show that the general solution to the nonhomogeneous problem is given by
!
√
4mk − b2
F0
(−b/m)t
x(t) = Ae
sin
t+φ + p
sin(γt + θ).
2m
(k − mγ 2 )2 + b2 γ 2
14.
RLC Series Circuit. In the study of an electrical circuit consisting of a resistor, capacitor,
inductor, and an electromotive force, we are led to an initial value problem of the form
q
dI
+ RI +
= E(t)
dt
C
q(0) = q0 , I(0) = I0 .
L
(1)
where L is the inductance in henrys, R is the resistance in ohms, C is the capacitance in
farads, E(t) is the electromotive force in volts, q(t) is the charge in coulombs on the capacitor
dq
is the current in amperes.
at time t, and I =
dt
Find the current at time t if the charge on the capacitor is initially zero, the initial current is
0, L = 10henrys, R = 20ohms, C = 6260−1 farads and E(t) = 100volts.
Hint: Differentiate both sides of the differential equation to obtain a homogeneous linear
second order equation for I(t). Then use equation (1) to determine dI
dt at t = 0.
15.
An RLC series circuit has an elctromotive force given by E(t) = sin 100t volts, a resistor of
0.02 ohms, an inductor of 0.001 henrys, and a capacitor of 2 farads. If the initial current and
the initial charge on the capacitor are both zero, determine the current in the circuit for t > 0.
4
SOLUTIONS TO TUTORIAL 2
1. a. y = c1 ex/2 + c2 e−4x .
b. y = (c1 + c2 x)ex/2 .
√
√
d. y = e2x (c1 cos 2 3x + c2 sin 2 3x).
c. y = c1 cos x + c2 sin x.
2. a. y = 3 − e−x .
√
√
1 c. y = √ e(2+ 2)x − e(2− 2)x .
2 2
x
x
e. y = ex/2 (sin − cos ).
2
2
4 x 1 3x
e − e .
3
3
d. y = (1 − 3x)ex .
b. y =
1
3. a. y = e5x − xe5x .
b. y = sin 2x.
2
e2x − e−3x
2 3x/2 2 −2x
c. y = −3/2
.
d.
y
=
e
− e
.
7
7
3e
+ 2e−2
t
t
4. a. y = c1 cos √ + c2 sin √ + 3t2 + 10t − 12.
2 √
2
!
√
35
35
x/2
x + c2 sin
x + cos 3x.
b. y = e
c1 cos
2
2
√
√ !
3
3
−x/2
x + c2 sin
x + sin 2x.
c. y = e
c1 cos
2
2
3
1
d. θ = c1 e2r + c2 e3r + er + rer .
4
2
1
1
−t
t
e. θ = c1 e + c2 e − t sin t − t cos t.
2
2
f. y = (c1 + c2 x)ex + 4x2 ex .
g. y = c1 ex + c2 e−x + 11x − 1.
1
1
h. y = c1 cos 2θ + c2 sin 2θ − θ cos 2θ − cos θ.
4
3
i. y = e−θ (c1 cos θ + c2 sin θ) + θe−θ cos θ.
√
j. y = c1 e
3x
√
+ c2 e−
3x
1
2 1
− x2 − + ex .
3
5 2
1
k. y = (c1 + c2 t)e2t + t3 e2t .
6
5
11 3x
x
−x
l. y = c1 e + c2 e +
x−
e .
8
32
5. a. y = ex − 1.
1 −4x 7 3x
1
1
1
c. y =
e
+ e − ex − e2x + .
60
6
10
6
12
b. y = − cos x + sin x + e−x .
7 θ 3 −θ 1
1
d. y =
e + e − sin θ − e2θ .
12
4
2
3
6. a. yp = (Ax2 + Bx + C)e2x
b. yp = A cos x + B sin x + C cos 2x + D sin 2x.
c. yp = Ae5x + (Bx + C)(D cos 3x + E sin 3x).
d. yp = x2 (Ax2 + Bx + C)e2x .
e. yp = Axe−3x + (Bx + C)(D sin 3x + E cos 3x) + F sin 2x + G cos 2x.
5
7. a. y = c1 cos x + c2 sin x + x sin x + cos x ln | cos x|.
1 1
b. y = c1 cos x + c2 sin x + − cos 2x.
2 6
1
1
c. y = c1 ex + c2 e−x + xex − xe−x .
4
4
d. y = c1 cos x + c2 sin x + x cos x + sin x ln | sec x|.
ex
1
− cos 2x ln | sec 2x + tan 2x|.
5
2
f. y = c1 e−x + c2 e−2x + (e−x + e−2x ) ln(1 + ex ).
e. y = c1 cos 2x + c2 sin 2x −
g. y = c1 e2x + c2 e−x − e−2x sin e2x .
1
h. y = c1 ex + c2 xex − [ex ln(x2 + 1) + xex tan−1 x].
2
1
3
−x
−x
i. y = c1 e + c2 xe + x2 e−x ln 2x − x2 e−x .
2
4
j. y = c1 cos x + c2 sin x − x2 + 3 + 3x sin x + 3 cos x ln | cos x|.
1
k. y = c1 e2t + c2 e−t + e3t .
4
1
l. y = c1 cos 2x + c2 sin 2x + (cos 2x ln | csc 2x + cot 2x| − 1).
4
8. a. y = 0.3 cos 5t − 0.02 sin 5t.
b. 0.8Hz
9. a. y = e−3t (0.3 cos 4t + 0.2 sin 4t).
b. 5/2π.
c. Exponential factor (damping factor)
√
√
3
10. y = e−3x cos 7x + √ sin 7x .
7
1
11. y = cos 3t + t sin t.
2
−F0
F0
12. y =
cos ωt +
sin γt.
m(ω 2 − γ 2 )
m(ω 2 − γ 2 )
2 −t
e sin 25t.
5
105
95
20
95
cos 20t −
sin 20t −
cos 100t +
sin 100t.
15. I = e−10t
9.425
18.85
9.425
9.425
14. I =