# Linear Simultaneous Equations

Linear simultaneous differential equations

## Definition

A**simultaneous differential equation**is one of the mathematical equations for an indefinite function of one or more than one variables that relate the values of the function. Differentiation of an equation in various orders. Differential equations play an important function in engineering, physics, economics, and other disciplines.This analysis concentrates on linear equations with Constant Coefficients.

## Using The D Operator

The

For example

**D operator**is a linear operator applied to functions and which is defined as .For example

## Using Laplace Transform

**Laplace transform**or the Laplace operator is a linear operator applied to functions and which is defined as where

Example:

##### Example - Apply Laplace

Problem

Apply the Laplace Transform and find and :
Give that at

Workings

The equations may be written as:
Hence
Therefore
Then we multiply the equation (3) with :
We subtract equation (4) from equation (5) :

2\mathhf{L\{\frac{dx}{dt}\}} + \mathhf{L\{\frac{dy}{dt}\}} + \mathhf{L\{90\,x\}} = \mathhf{L\{45\}}

\mathhf{L\{\frac{dx}{dt}\}} + 2\mathhf{L\{\frac{dy}{dt}\}} + \mathhf{L\{240\,y\}} = 0

Solution

Therefore
From equation (4) we get :