Topic For Advance Math prepared by our Professor

1. The Inverse Transforms
Felix V. Garde, Jr.
Felix V. Garde, Jr. The Inverse Transforms 1 / 10

2. Definition
If F(t) is of class A, from
L{F(t)} = f(s)
then the inverse Laplace transform of f(s) is defined as
F(t) = L−1
{f(s)}
Theorem
If c1 and c2 are constants,
L−1
{c1f1(s) + c2f2(s)} = L−1
{c1f1(s)} + L−1
{c2f2(s)}
Felix V. Garde, Jr. The Inverse Transforms 2 / 10

3. Laplace of elementary functions
L{ekt
} =
1
s − k
, s > k
L{1} =
1
s
, s > 0
L{tn
} =
n!
sn+1
, s > 0
L{sin kt} =
k
s2 + k2
, s > 0
L{cos kt} =
s
s2 + k2
, s > 0
L−1
{f(s − a)} = eat
L−1
{f(s)}
Felix V. Garde, Jr. The Inverse Transforms 3 / 10

4. Examples
Obtain L−1
{f(s)} from the given f(s).
•
1
s2 + 2s + 10
•
s
s2 + 6s + 13
•
2s + 3
(s + 4)3
•
5s − 2
s2(s + 2)(s − 1)
• Show that
L−1 1
(s + a)2 + b2
=
1
b
e−bt
sin bt
• For a > 0, show that L−1
{f(s)} = F(t) it follows that
L−1
{f(as)} =
1
a
F
t
a
Felix V. Garde, Jr. The Inverse Transforms 4 / 10

5. Examples
Initial value problems: Solve the following using Laplace transform
method.
• y = et
; y(0) = 2
• y + y − 2y = −4; y(0) = 2, y (0) = 3
• x (t) + 3x (t) + 2x(t) = 4t2
; x(0) = 0, x (0) = 0
• x (t) − 4x (t) + 4x(t) = e2t
; x (0) = 0, x(1) = 0
Felix V. Garde, Jr. The Inverse Transforms 5 / 10

6. Definition
The Unit Step Function
The function u(t) is defined as
u(t) =
0 t < 0
1 t ≥ 0
and u(t − a) is
u(t − a) =
0 t < a
1 t ≥ a
Felix V. Garde, Jr. The Inverse Transforms 6 / 10

7. Definition
The Unit Step Function
(a) (b)
Graphs of unit step functions. (a) u(t) (b) u(t − a)
Felix V. Garde, Jr. The Inverse Transforms 7 / 10

8. Examples
Sketch the graph of the given functions for t ≥ 0.
• α(t − 1) + 2α(t − 2) − 3α(t − 4)
• (t − 3)α(t − 3)
• t2 − (t − 1)2α(t − 1)
Felix V. Garde, Jr. The Inverse Transforms 8 / 10

9. Definition
The Laplace transform of α(t − c)F(t − c)
L{α(t − c)F(t − c)} =
∞
0
e−st
α(t − c)F(t − c)dt
is
L{α(t − c)F(t − c)} = e−cs
L{F(t)} = e−cs
f(s)
Theorem
If L−1{f(s)} = F(t), if c ≥ 0, and if F(t) be assigned values for
−c ≤ t < 0,
L−1
{e−cs
f(s)} = F(t − c)α(t − c)
Felix V. Garde, Jr. The Inverse Transforms 9 / 10

10. Examples
Express F(t) in terms of the α function and find L{F(t)} .
•
F(t) =
3 0 < t < 1
t t > 1
•
F(t) =



t2
0 < t < 1
3 1 < t < 2
0 t > 2
• Evaluate L−1 e−4s
(s + 2)3
• x (t) + x(t) = F(t); x(0), x (0) = 0, in which
F(t) =
4 0 ≤ t ≤ 2
t + 2 t > 2
Felix V. Garde, Jr. The Inverse Transforms 10 / 10

11. A Convolution Theorem
Given
L−1
{f(s)} = F(t), L−1
{g(s)} = G(t)
are functions of class A, then
L−1
{f(s)g(s)} =
t
0
G(β) F(t − β)dβ
or
L−1
{f(s)g(s)} =
t
0
F(β) G(t − β)dβ
Felix V. Garde, Jr. The Inverse Transforms 11 / 10

12. Examples
Find the Laplace transform of the given convolution integral
•
t
0
(t − β) sin 3βdβ
•
t
0
(t − β)3
eβ
dβ
Find the inverse Laplace transform of the given f(s) using the convolution
theorem.
•
1
s(s2 + k2)
•
1
s(s + 2)
• Solve the problem
x (t) + 2x (t) + x(t) = F(t); x(0) = 0, x (0) = 0
Felix V. Garde, Jr. The Inverse Transforms 12 / 10