Laplace Transform : Why is it Powerful?

Laplace Transform is a very interesting tool of Mathematics, having wide range of applications in System Modeling, Nuclear Physics, Computer Science and many other technical technical disciplines of engineering.
In the very root, Laplace Transform (L.T.) is a concept which is used to solve certain kind of differential equations which, in-turn, facilitate its applications stated above.

Laplace Transform Equations

Let there be a function F(t) with ‘t’ as independent variable. Then the L.T. of the function F(t) will be

Laplace Transform Equation

Where f(p) is another function with ‘p’ as its independent variable.

In order to supplement the process of solving certain differential equations with given constants, we have a reciprocating concept of L.T. called Inverse Laplace Transform.

Inverse Laplace Transform

Inverse Laplace Transform Equation

where F(t) and f(p) are different functions of ‘t’ and ‘p’ respectively.

What is Laplace Transform and How Does it Work?

L.T. trace back its history from a concept of power series method of solving differential equations.

Power Series Method

A Power Series is an infinite series of the form

Laplace Transform Power Series

We know that a differential equation is an equation containing a function, its variables and functional derivatives of the same.
The solution of the differential equation is said to be the equation of the function itself, without any derivatives.

Now in power series method we assume that the function ‘y’ is equal to infinite power series “summation”
Then for the assumed equation we find the required coefficients satisfying the differential equation.
Then putting everything in the series we get the solution.

For example: Solving by Power Series Method the differential equation (1-x)(dy/dx) = y

Laplace Transform Equation

Getting back to the concept, L.T. is a continuous form of power series solution.

Laplace Transform Equation Image


Laplace Transform Properties

Here we have some properties of L.T.

Laplace Transform Properties

Applications of Laplace Transform

Theorem used in solving different differential equation by L.T. Method

  1. Differentiation Theorem
Inverse Laplace Transform Equation 2

Steps for solving linear differential equations with constant coefficients.

  1. Taking L.T. of the given differential equation
  2. Using Differential Theorem reducing the transformed differential equation to zero order.
  3. Rearranging the equation like L(y)=f(p) ; where f(p) is the function of p.
  4. Taking Inverse L.P. Transform of the function f(p) Y=L’f(p)
  5. Y is the solution of the given differential equation.
Laplace Transform Equation Image