# 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: