Section 4.1 Laplace transformation

Objective:

1. Definition of Laplace transform

2. Motivation of learning Laplace transformation

3. Proof of basic Laplace transformation formula

In this chapter, we only have one main theme, Laplace transformations. The motivation is to solve more ODE questions that presents the real life situations that cannot be solved in previous chapters. Recall that we need continuous coefficient functions in order to have solutions exists. Here, Laplace transformation will give us a powerful tool to solve the ODE even the coefficient functions are not continuous, for example the impulse functions, the special functions that have very short period of nonzero values. 

Definition: The Laplace Transform of $f$ }is defined as an integral transform:

$$L\{f(t)\}=F(s)={\int }_0^{\mathrm{\infty }}{e}^{-st}f(t)dt.$$

Note that in general, an improper integral may not convergent, hence there are conditions to apply in order to have the improper integral be finite. 

Theorem: Suppose that $f$ is a function for which the following hold:

(1) $f$ is piecewise continuous on $[0,b]$ for all $b>0$. 

(2) $|f(t)|\le K{e}^{at}$ when $t>M,$ for some constants $a,$ $K$, $M$, with $K,M>0$. Then the Laplace Transform of $f$ exists for $s>a$ and 

$$L\{f(t)\}=F(s)={\int }_0^{\mathrm{\infty }}{e}^{-st}f(t)dt$$

 is finite. 

Example 1: Find Laplace Transform of $1$ and $t^3$. 

Exercise 1: Find Laplace Transform of $t$ and $t^2$. 

Example 2: Find Laplace Transform of $\text{sin}(at)$ where $a$ is a constant. 

Exercise 2: Find Laplace Transform of $\text{cos}(at)$ where $a$ is a constant. 

Example 3: Find Laplace Transform of $e^{at}$ where $a$ is a constant. 

Exercise 3:  Find Laplace Transform of $e^{3t}$.

Example 4: Find Laplace Transform of $e^{2t}-5t$.

Exercise 4: Find Laplace Transform of $3+2t$.

Theorem: (Linearity) Suppose $f$ and $g$ are functions whose Laplace transforms exist for $s>a_1$ and $s>a_2$ respectively. Then, for $s$ greater than the maximum of $a_1$ and $a_2$, the Laplace transform of $c_{1}f(t)+c_{2}g(t)$ exists.

That is,

$$L\{c_{1}f(t)+c_{2}g(t)\}={\int }_0^{\mathrm{\infty }}{e}^{-st}[c_{1}f(t)+c_{2}g(t)]dt$$

 is finite and 

$$L\{c_{1}f(t)+c_{2}g(t)\}=c_{1}L\{f(t)\}+c_{2}L\{g(t)\}.$$

Group Work

Find Laplace transform of the given functions.

1. $2+3t^2-e^{3t}$

2. $5\text{cos}(4t)+t^3+4e^{5t}$

3. $5\text{sin}(2t)-t^4+2e^{-4t}+6$

4. $3\text{sin}(4t)-\text{cos}(2t)+2e^{6t}$