PURE MATHEMATICS - Integration

 

Differential Equations

 

 

introduction

variables separable

the integrating factor

 

 

 

Introduction

 

All equations with derivatives of a variable w.r.t. another are called 'differential equations'.

 

A first order differential equation contains a first derivative eg dy/dx.

 

It might not be appreciated, but ALL integrals are derived from original 'first-order' differential equations.

 

 

Example:

     Write y as a simple function of x.

 

 

(where C is the constant of integration)

 

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First Order with 'Variables Separable'

 

Solution is by collecting all the 'y' terms on one side, all the 'x' terms on the other and integrating each expression independently.

 

                                               

 

              rearranging                                    

        

integrating both sides                         

 

                            

 

 

Example #1

rearranging                                                          

 

integrating both sides

 

        multiplying by 6

 

            dividing by 9

 

Note how the constant of integration C changes its value.

 

 

Example #2

                                   

 

rearranging          

 

integrating both sides       

 

               multiplying both siides by 2y                                        

 

 

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First Order 'linear' differential equations

 

By definition 'linear' differential equation have the form:

 

 

Dividing by f(x) to make the coefficient of dy/dx equal to '1', the equation becomes:

 

 

(where P and Q are functions of x, and only x)

 

The key to solving these types of problem is to choose a multiplying factor(sometimes called an 'integrating factor').

 

This is to make the LHS of the equation appear like a result from the Product Rule.

 

 

Example      Find the P and Q functions and for the following differential equation, expressing it in terms of x and y only.

 

rearranging,        

 

                               cancelling x's                        (i                

 

recalling that,         2nd

 

               

 

 

multiplying equation (i by to make the LHS like a Product Rule result,

 

product rule

where and

 

                                                     (ii

 

 

 

 

     from the equality (ii                                              

 

integrating                  

                                           

 

 

 

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