SCEE08009
Engineering Mathematics 2A
Exercise ODE 1
A number of the questions are taken from or similar to Exercises in Modern Engineering Mathematics and Advanced Modern Engineering Mathematics. You will get the
best from this tutorial by working through these examples before the tutorial session. If
you are having difficulties with these questions, practice using the remaining questions
from these exercises.
1. Which of the following sets are linearly independent and which are linearly dependent?
(a) f = {1, t, t2 , t3 , t4 , t5 , t6 } = {f1 , f2 , f3 , f4 , f5 , f6 , f7 }
(b) f = {1 + t, t2 , t2 − t, 1 − t2 } = {f1 , f2 , f3 , f4 }
(c) f = {sin t, cos t, sin t − cos t, 2 sin t + cos t, 2 sin t − cos t} = {f1 , f2 , f3 , f4 , f5 }
(d) f = {1 − 2t2 , t − 3t3 , 3t2 + 4t3 , t3 } = {f1 , f2 , f3 , f4 }
(e) f = {et , e2t − et , e3t − e2t , e2t } = {f1 , f2 , f3 , f4 }
(f) f = {1 − 2t2 , t − 3t3 , 2t2 − 4t4 , 4t3 } = {f1 , f2 , f3 , f4 }
2. For each of the following differential equations, write down the differential operator,
L, that will allow the equation to be expressed in the form L[x(t)] = 0.
(a)
dx
− kx = 0
dt
(b)
d2 x
d3 x
+
(sin
t)
+ 4t2 x = 0
dt3
dt2
(c)
d
dt
d
2 dx
t
= t (xt)
dt
dt
d
dt
(d)
1d 2
(t x)
t dt
= xt.
3. Determine which members of the given sets s are solutions of the differential equation.
Hence write down the general solution of the differential equation
Remark: The general solution of a homogeneous differential equation is equal to its
complementary solution because its particular integral is equal to zero.
(a)
d2 x
− p2 x = 0
2
dt
s = ept , e−pt , cos pt, sin pt
SCEE08009
Engineering Mathematics 2A
Exercise ODE 1
(b)
d2 x
dx
=0
+2
2
dt
dt
s = cos 2t, sin 2t, e−2t , e2t , t2 , t, 1
(c)
d3 x d2 x dx
+x=0
− 2 −
dt3
dt
dt
s = et , e−t , e2t , e−2t , tet , te−t , te2t , te−2t
4. Solve the following initial value problems:
(a)
d2 x
dx
dx
2 2 −2
+ 3x = 0, x(0) = 1,
= 0.
dt
dt
dt t=0
(b)
d2 x
d3 x
d2 x
dx
dx
=
0,
−
6
+
11
−
6x
=
0,
x(0)
=
1,
= 1.
dt3
dt2
dt
dt t=0
dt2 t=0
(c)
d2 x
d2 x
dx
dx
d3 x
=
1,
+
6
+
12
+
8x
=
0,
x(1)
=
1,
= 0.
dt3
dt2
dt
dt t=1
dt2 t=1
5. A initial value problem (IVP) comprises
−
d2 x
dx
−4
− 8x = 0
2
dt
dt
together with the initial conditions x(0) = 1 and ẋ(0) = 0
(a) Derive the complementary function and show that it is
x(t) = e−2t (A cos 2t + B sin 2t) .
(b) Given that the complementary solution to the ODE is
x(t) = e−2t (A cos 2t + B sin 2t) ,
use the boundary conditions, x(0) = 1 and ẋ(0) = 0, find the particular solution
to the given IVP.
6. Analyze the the linear dependence/independence of the following set of functions
using the Wronskian.
(a) f = {ex , e−3x } = {f1 , f2 } for x ∈ R.
(b) f = {t2 + 1, 2t2 + 2} = {f1 , f2 } for t ∈ R
Page 2