LECTURE 23:THE LAPLACE TRANSFORM AND SYSTEMS OF ODES
MINGFENG ZHAO
November 02, 2015
The Laplace transform
Definition 1. Let f (t) be a function on [0, ∞), then
I. The Laplace transform of f , denoted by L[f ](s), is defined as:
∫ ∞
L[f (t)](s) =
f (t)e−st dt, for all s > 0.
0
II. If F (s) = L[f ](s), the inverse Laplace transform of F , denoted by L−1 [F ](t), is defined as:
L−1 [F (s)](t) = f (t),
for all t > 0.
Definition 2. Let a be a real constant, and the Heaviside function is defined as:

 1, if t ≥ a,
u(t − a) =
 0, if x < a.
Remark 1. Notice that when a = 0, we know that u(t − 0) = 1 for all t ≥ 0; when a = ∞, we define u(t − ∞) := 0 for
all t ≥ 0.
Definition 3. Let f (t) and g(t) be two functions on [0, ∞), the convolution of f and g is defined as:
∫ t
(f ∗ g)(t) =
f (τ )g(t − τ ) dτ.
0
Definition 4 (Dirac’s delta function). For any continuous function f (t) on (−∞, ∞), we have
∫ ∞
δ(t)f (t) dt = f (0).
−∞
Proposition 1. The followings hold:
I. Transforms of derivatives:
L[f ′ ](s)
=
sL[f ](s) − f (0)
L[f ′′ ](s)
=
s2 L[f ](s) − sf (0) − f ′ (0).
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2
MINGFENG ZHAO
II. First Shifting Property:
[
]
L e−at f (t) (s) =
L−1 [F (s + a)](t) =
L[f ](s + a)
e−at L−1 [F (s)](t).
III. Second Shifting Property:
e−as L[f (t)](s)
L[u(t − a)f (t − a)](s) =
[
]
L−1 e−as F (s) =
u(t − a)L−1 [F (s)](t − a).
IV. Transform of Integrals:
[∫
t
L
]
f (τ ) dτ (s)
0
L−1
[
]
F (s)
(t)
s
=
=
L[f (t)](s)
s
∫ t
L−1 [F (s)](τ ) dτ
0
V. Transform of Convolution:
L[(f ∗ g)(t)](s) =
L−1 [F (s) · G(s)] (t) =
L[f (t)](s) · L[g(t)](s)
L−1 [F (s)] ∗ L−1 [G(s)](t).
VI. Transform of Dirac delta:
L[δ(t − a)](s) =
L−1 [e−as ](t) =
e−as
δ(t − a).
Remark 2. In practice, we can use the second shifting property in the following way:
L[u(t − a)f (t)](s) = e−as L[f (t + a)](s).
Example 1. Solve y ′′ + 4y ′ + 5y = δ(t − 3), y(0) = 1, y ′ (0) = 0.
Let Y (s) = L[y(t)], apply the Laplace transform on the both sides of y ′′ + 4y ′ + 5y = δ(t − 3), then
L[y ′′ ] + 4L[y ′ ] + 5L[y] = L[δ(t − 3)].
By the transform of derivatives, we have
L[y ′ ] = sL[y] − y(0) = sY (s) − 1,
and L[y ′′ ] = s2 L[y] − sy(0) − y ′ (0) = s2 Y (s) − s.
LECTURE 23:THE LAPLACE TRANSFORM AND SYSTEMS OF ODES
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By looking the table, we have L[δ(t − 3)] = e−3s , so we get
s2 Y (s) − s + 4[sY (s) − 1] + 5Y (s) = e−3s .
Then
Y (s)
=
=
=
e−3s + s + 4
s2 + 4s + 5
s+4
e−3s
+ 2
2
s + 4s + 5 s + 4s + 5
s+2
2
e−3s
+
+
.
2
2
(s + 2) + 1 (s + 2) + 1 (s + 2)2 + 1
By the first shifting property and looking up the table, we have
]
]
[
[
s+2
1
−2t
−1
L−1
(t)
=
e
cos(t),
and
L
(t) = e−2t sin(t).
(s + 2)2 + 1
(s + 2)2 + 1
By the Second Shifting Property, we have
[
]
[
]
e−3s
1
−1
−1
L
= u(t − 3) · L
(t − 3) = u(t − 3)e−2(t−3) sin(t − 3).
(s + 2)2 + 1
(s + 2)2 + 1
Then we get
y(t) = L−1 [Y (s)] = e−2t cos(t) + 2e−2t sin(t) + u(t − 3)e−2(t−3) sin(t − 3).
Therefore, the solution to y ′′ + 4y ′ + 5y = δ(t − 3), y(0) = 1, y ′ (0) = 0 is:
y(t) = e−2t cos(t) + 2e−2t sin(t) + u(t − 3)e−2(t−3) sin(t − 3) .
Introduction to systems of ODEs
In this course, we only study the systems of the first order differential equations which has the form:

 x′ (t) = f1 (t, x1 , x2 )
1
 x′ (x) = f (t, x , x ).
2
2
1
2
A second order differential equation can be viewed as a system of the first order differential equations, e.g., x′′ =
f (t, x, x′ ). Let x1 (t) = x(t) and x2 (t) = x′ (t), then

 x′ (t) = x′ (t) = x2
1
 x′ (t) = x′′ (t) = f (t, x, x′ ) = f (t, x , x ).
1
2
2
Remark 3. In general, an n-th order differential equation can be transformed into a first order system with n unknown
functions.
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MINGFENG ZHAO
In this course, we only study the linear system in the following form:

 x′ = ax1 + bx2 + f1 (t)
1
 x′ = cx + dx + f (t)
1
2

Let ⃗x(t) = 

x1 (t)
,

A=
x2 (t)

a
b
c
d
2
2

,
and f⃗(t) = 

f1 (t)
, then we have ⃗x′ = A⃗x + f⃗(t) . The matrix A is
f2 (t)
⃗
called the coefficient matrix of the system, and f (t) is called the non-homogeneous term of the system. If f⃗(t) ≡ 0, we
say ⃗x′ = A⃗x is homogeneous.
Theorem 1. Let ⃗xc (t) be the general solution to the homogeneous system ⃗x′ = A⃗x, and ⃗xp (t) be a particular solution
to ⃗x′ = A⃗x + f⃗(t), then the general solution to ⃗x′ = A⃗x + f⃗(t) is ⃗x(t) = ⃗xc (t) + ⃗xp (t).
Method of Elimination
Let’s solve the system:
(1)
x′1
= ax1 + bx2 + f1 (t)
(2)
x′2
= cx1 + dx2 + f2 (t).
Take the derivative of the first equation (1), then
x′′1 = ax′1 + bx′2 + f1′ (t).
(3)
By (2) and plug x′2 into (3), then
x′′1
(4)
=
ax′1 + b (cx1 + dx2 + f2 (t)) + f1′ (t)
=
ax′1 + bcx1 + bdx2 + bf2 (t) + f1′ (t)
By (1), we know that bx2 = x′1 − ax1 − f1 (t), plug bx2 into (4), we get
x′′1
= ax′1 + bcx1 + d (x′1 − ax1 − f1 (t)) + bf2 (t) + f1′ (t)
= (a + d)x′1 − (ad − bc)x1 + f1′ (t) − df1 (t) + bf2 (t).
That is, we have
(5)
x′′1 − (a + d)x′1 + (ad − bc)x1 = f1′ (t) − df1 (t) + bf2 (t).
LECTURE 23:THE LAPLACE TRANSFORM AND SYSTEMS OF ODES
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The equation (5) is a constant coefficient of second order equation, we can use undetermined coefficients or variation
of parameters to solve x1 from (5), that is, there exists some functions y1 (t), y2 (t)(t) and x1,p (t) such that
x1 (t) = C1 y1 (t) + C2 y2 (t) + x1,p (t).
Now by taking the derivative of the second equation (2), and by using the same argument as before, we know that
x2 satisfies:
(6)
x′′2 − (a + d)x′2 + (ad − bc)x2 = f2′ (t) − bf2 (t) + cf1 (t).
The equation (6) is a constant coefficient of second order equation, we can use undetermined coefficients or variation
of parameters to solve x2 from (6). Since the homogeneous equations for (5) and (6) are the same, then there exists
some function x2,p (t) such that
x2 (t) = D1 y1 (t) + D2 y2 (t) + x2,p (t).
Then plug x1 (t) and x2 (t) into (1) and (2) to find the relations of C1 , C2 , D1 and D2 . In the end, we only have two
free constants.
Problems you can do:
Lebl’s Book [2]: All exercises on Page 88 and Page 89.
Braun’s Book [1]: Read all materials in Section 3.1, Section 3.2, Section 3.3, Section 3.5, Section 3.6 and
Section 3.7.
References
[1] Martin Braun. Differential Equations and Their Applications: An Introduction to Applied Mathematics. Springer, 1992.
[2] Jiri Lebl. Notes on Diffy Qs: Differential Equations for Engineers. Createspace, 2014.
Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, B.C.
Canada V6T 1Z2
E-mail address: mingfeng@math.ubc.ca