Laplace Transformation

Laplace transformations are a mathematical technique used to solve differential equations. Numerous mathematical problems are solved using transformations. The idea is to transform a difficult problem into another form that is easier to solve. Once the problem is solved, the inverse transform can be used to solve the original problem. 

The Laplace Transformation of a function f(t) can be converted into a function f(s) using the following equation:

Example 1: Find the Laplace Transform of f(t) = 1

Solution:

Example 2: Find the Laplace Transform of f(t) = e^at

Solution:

Example 3: Find the Laplace Transform of f(t) = sin (wt) and f(t) = cos (wt) 

Solution:

Example 4: Find the Laplace Transform of f(t) = cosh (wt) and f(t) = sinh (wt)

Solution: 

Below is a summary table for common Laplace Transformations

Shifting Theorem

The common expression is as follows:

Example 5: Find the Laplace Transform of f(t) = e^at cos (bt)

Solution:

Example 6: Find the Laplace Transform of f(t) = sin (3t).e^5t 

Laplace Transforms for Derivatives

The following are the general equations for different ordered differential equations:

Example: Find the Laplace Transform of the function x(t) which satisfy the following differential equation and initial condition:

Solution:

Take the Laplace transform of both sides of the equation.

Laplace Transforms for Integrals 

The general equation for Laplace transforms of integrals is as follows:

Example: Find x(s) for the following equations

Solution