# 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.

The concept is simple. Instead of calculating the solution of a Differential Equation directly by the usual methods, apply Laplace Transform, convert it to algebraic form, solve it and then inversely convert the solution to the form it should be. Figure 1: Solving Differential Equations

## Examples

*For simplicity, all the initial conditions are considered zero.

### Example 1:      ### Example 2:       ### Example 3:      Now we will calculate the coefficients of the partial fractions:    Substituting the coefficients to the partial fractions:   ### Example 4:      Now we will calculate the coefficients of the partial fractions:  The coefficients a and c will be calculated using the equating coefficients method:   Equating the partial fractions polynomial with the initial polynomial we have: The nominators must be equal, so we have the equations system: Substituting the coefficients to the partial fractions we have:   