1. Week 8 Laplace transformation The basics The Shifting Theorems
Applications of the LT to ODEs

2. 1. The basics
۞ Consider f(t) defined for t ≥ 0. Then, if the integral
exists, it is called the Laplace transform of f(t).
Alternative notation:
Example 2:

3. (integrate by parts n times)
Laplace transformation is a linear – hence,

4. Example 3:
Theorem 1: Existence of Laplace transforms
Let f(t) be a piece-wise continuous function for t ≥ 0 and
where M and a are positve constants. Then L[f(t)] exists for s ≥ a.

5. If we are given a Laplace transform F(s), we can find the original function f(t) through the inverse LT,
Comment:
Since the forward LT doesn’t ‘know’ how f(t) behaves at t < 0, the inverse LT can’t restore this part of f(t).
There are two ways to find inverse transforms: by using the table of transforms and by evaluating a certain complex integral.
Example 4:
Find

6. The inverse LT is linear, i.e.
Example 5:
Find
۞ The Heaviside step function, u(t – a), is
hence...

8. Oliver Heaviside (1850 –1925) was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques for solving differential equations (later found to be equivalent to the Laplace transformation), and predicted the existence of the “Heaviside layer” in the ionosphere. He didn’t go to university and his only paid employment was a telegraph operator job (while he was years old).
In later years his behaviour became quite eccentric: he would sign letters with the initials “W.O.R.M.” after his name, used granite blocks for furniture, and reportedly started painting his fingernails pink.
For most of his life, Heaviside was at odds with the scientific establishment. Most of his recognition was gained posthumously.

10. Comment:
Observe that
i.e. the inverse transforms of two different functions coincide... how come?
2. The Shifting Theorems
Theorem 2: 1st Shifting Theorem
If L-1[F(s)] = f(t), then
Proof:
By definition of the LT...

11. hence,
hence,
Apply L-1 to this equality to obtain the desired result. █
Example 6:
Find

12. Solution:
F(s) = (s – 2)–4 is not in the table, but, using the 1st Shifting Theorem with a = –2, we can reduce the problem to F(s) = s–4:
Example 7:
Find
The answer:

13. Theorem 3: 2nd Shifting Theorem
If L-1[F(s)] = f(t), then
Alternative formulation:
Example 8:
Find
Solution:
Use the 2nd Shifting Theorem with a = 6 and F(s) = s–4...