# Solving Differential Equations Using Laplace Transform

## Introduction

Solving Differential Equations can become a very difficult task for large and complicated equations. Also solving systems of Differential Equations is a very difficult task. However, there is a tool that simplifies the calculations to something that can be relatively easy to calculate. This tool is the Laplace Transform.

# Defintion of Integration - Indefinite Integral

A function F(x) is called antiderivation of f(x) when it is:

F(x) is also called integral of f(x). The procedure of determining the integral is called integration.

Since the derivation of a constant C is zero, then for a function there are an infinite number of integrals:

# Multiple Integrals

### Double Integral

A double integral consists of two successive integrations of a function of two variables. We calculate each integration independently of the other.

We initially calculate the integration inside the brackets, considering x as constant and then we calculate the resulting function with respect to x. Geometrically the result represents the volume between xy plane and the surface f(x,y) in the three dimensional Cartesian System inside the limits of the integration.

# Definitions

The hyperbolic functions are defined using Euler’s number e=2.71. They are very frequently part of the equations used in Electrical and Electronic Engineering. The hyperbolic functions are:

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