
Laplace Transform

Introduction
Consider following equations:
Clearly, first equation is algebraic and second equation is differential. It takes a lot of efforts to solve the differential equation, when compared to algebraic equation.
The Laplace transform is a tool, named after French mathematician PierreSimon Laplace, which converts a differential equation into an algebraic equation.
Note that the solution will be a function instead of a number.

Definition
Let be a realvalued function, defined for . The Laplace transform of is defined as
Thus, it transforms into another function (of a complex variable) .

When Does Laplace Transform Exist?
Note : Important from a math. perspective, can be left out by engineering students. Nonetheless, it’s necessary to know this.
Let there be a constant such that is bounded as . In other words,
Then, is said to be of exponential order .
If is piecewise continuous in every finite interval of and is of exponential order , then Laplace transform of , i.e. exists for all .

Linearity (Superposition Principle) of Laplace Transform
If and are constants and and are 2 functions with existence of Laplace transforms, then,

Laplace Transforms of Standard Functions

First Shifting Theorem
If , then,

Second Shifting Theorem
If and and , then,

Change of Scale Theorem
If , then,

Laplace Transform of Derivatives of Functions, Roughly corresponds to multiplication
If , then,
$latex{\mathcal {L} [f'(t)] =s F(s) – f(0)}&s=2$
Form an expression for .

Laplace Transform of Derivatives of Functions, Roughly corresponds to division
If , then,

Laplace Transform of and , Roughly corresponds to differentiation and integration

Convolution
The convolution of 2 functions and is given by
It is commutative, associative and distributive over addition.
Laplace transform of convolution is given by
It can be shown that .

Initial Value Theorem,
If , then,

Final Value Theorem,
If , then,

Inverse Laplace Transform
This is an operation, which gets us back to the original function. In other words, if is the Laplace transform of , then the inverse Laplace transform is defined as
It is a linear transform.

Formulas
Having developed the formulas for Laplace transform, we can easily get the following :

Theorems
The theorems discussed for Laplace transform are presented here in terms of inverse Laplace transform.
Useful when has logarithmic or inverse trigonometric functions :
Useful when has a power of in the denominator :
Convolution Theorem :

Partial Fractions
The method of partial fractions greatly simplifies by splitting the function into fractions, whose denominators are components of the denominator of . Getting the coefficients of the fractions can take time, so try if the shifting theorem can be applied.

Solving ODEs using Laplace transform
We solved linear ODEs of higher order in 1st unit. They can also be solved using Laplace transforms. We first take the Laplace transform of the differential equation by using the properties above. This converts the ODE into an algebraic equation. We solve it to get . Taking the inverse Laplace transform of gives us the required function or the solution of the ODE.