Laplace Transformation Free Course

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Laplace transform is an integral transform that converts a function of a real variable, usually time, to a function of a complex variable or complex frequency. This transform is also used to analyze dynamical systems and simplify a differential equation into a simple algebraic expression. It changes one signal into another based on some fixed set of rules and equations. It is considered the best way to convert differential equations into algebraic equations and convolution into multiplication. It holds a major role in control system engineering. This transform has its applications in various applications in different types of science and engineering since it is a tool for solving differential equations.

Laplace transform is an integral transform that converts a function of a real variable, usually time, to a function of a complex variable or complex frequency. It is named after a great French mathematician, Pierre Simon De Laplace. It is considered a generalized Fourier series. It changes one signal into another based on some fixed set of rules and equations. It is considered the best way to convert an ordinary differential equation into algebraic equations and convolution into multiplication which aids in solving them faster and in a very easier approach.

Laplace transform is easy to apply to the problems and solve them if the constants are known. It is easy even to solve the complex nonhomogeneous differential equations.

Different Types of Laplace Transforms are:

• Bilateral Laplace Transform or Two-sided Laplace Transform
• Inverse Laplace transform
• Probability Theory

Laplace transform is used to simplify differential equations to an algebraic equation that is much easier to solve. It is restricted to solving complex differential equations with known constants. If there is a differential equation without known constants, then this method is totally useless. Some other method has to be applied to solve such problems.

It is always good to have a copy of Laplace transform tables. In that case, you do not need to memorize it, otherwise, you should learn them and register them if you want to ace Laplace transform.

Follow these simple steps to solve Laplace:

Method: Basics

Step 1: Substitute the function into the definition of Laplace transform.

Step 2: Evaluate the integrals through any means of your comfort.

Step 3: Evaluate the Laplace of the power function.

Method: Laplace Properties

Step 1: Find the Laplace of a function multiplied by eᵃᵗ.

Step 2: Find the Laplace of a function multiplied by tᵑ.

Step 3: Determine Laplace of a stretched function f(at)

Step 4: find the Laplace of a derivative F(t). 

Laplace of zero is computed as eᶺ(-infinity*t). This result shows 0. So, the Laplace of 0 is 0.