25 First-order differential equations PowerPoint

Programme 25: First-order differential equations

PROGRAMME 25

FIRST-ORDER

DIFFERENTIAL

EQUATIONS

STROUD Worked examples and exercises are in the text


Introduction

Formation of differential equations

Solution of differential equations

Bernoulli ’ s equation



Introduction





Introduction


A differential equation is a relationship between an independent variable x , a dependent variable y and one or more derivatives of y with respect to x .

The order of a differential equation is given by the highest derivative involved.

x dy dx

 y

2 

0 is an equation of the 1st order xy dx

2

 y

2 sin x



0 is an equation of the 2nd order d y

 y dy dx

3 dx

 e

4 x 

0 is an equation of the 3rd order



Introduction






Introduction







Differential equations may be formed from a consideration of the physical problems to which they refer. Mathematically, they can occur when arbitrary constants are eliminated from a given function. For example, let: y



A sin x



B cos so that dy dx



A cos x



B sin therefore dx

2

 

A sin x



B cos x

  y

That is dx

2

0




Here the given function had two arbitrary constants: y



A sin x



B cos x and the end result was a second order differential equation:

2 d y dx

2

0

In general an n th order differential equation will result from consideration of a function with n arbitrary constants.



Introduction






Introduction







Introduction

Direct integration

Separating the variables

Homogeneous equations – by substituting y = vx

Linear equations – use of integrating factor




Introduction

Solving a differential equation is the reverse process to the one just considered. To solve a differential equation a function has to be found for which the equation holds true.

The solution will contain a number of arbitrary constants – the number equalling the order of the differential equation.

In this Programme, first-order differential equations are considered.





If the differential equation to be solved can be arranged in the form: dy dx

 the solution can be found by direct integration. That is: y

 

( )





For example: so that: dy dx



3 x

2 

6 x



5 y

 

(3 x

2



6 x



5) dx

 x

3 

3 x

2 

5 x



C

This is the general solution (or primitive ) of the differential equation. If a value of y is given for a specific value of x then a value for C can be found.

This would then be a particular solution of the differential equation.





If a differential equation is of the form: dy



( ) dx ( )

Then, after some manipulation, the solution can be found by direct integration.

( )

 f x dx so



( )

 

( )





For example: so that:

That is:

Finally: dy

 dx y

2

 x

1

( y



1) dy



2 xdx

 y



1) dy

 

2 xdx y

2

+ y

+

C

1

= x

2 +

C

2

2 y

2

+ y

= x

2 +

C

2





In a homogeneous differential equation the total degree in x and y for the terms involved is the same.

For example, in the differential equation: dy

 x



3 y dx 2 x the terms in x and y are both of degree 1.

To solve this equation requires a change of variable using the equation:

y

= vx





To solve: let to yield: dy

 x



3 y dx 2 x y



( ) dy dx v x dv dx

and x



3 y



 v

2 x 2

That is: x dv dx



1



2 v which can now be solved using the separation of variables method.





Consider the equation: dy dx



5 y

 e

2 x

Multiply both sides by e 5 x to give: then:

That is:

ò

e

5 x dy dx

 e

5 x

5 y



5 x 2 x e e that is d dx

=

ò

e

7 x dx so that ye

5 x = e

7 x

+

C

7 y

= e

2 x

7

+

Ce

-

5 x

 e

7 x





The multiplicative factor e 5 x that permits the equation to be solved is called the integrating factor and the method of solution applies to equations of the form: dy dx



Py



Q e



Pdx

where is the integrating factor

The solution is then given as: y .IF

 

Q .IF

dx where IF

 e



Pdx



Introduction






Introduction







A Bernoulli equation is a differential equation of the form: dy dx



Py



Qy n

This is solved by:

(a) Divide both sides by y n to give: y

 n dy dx



Py

1

 n



Q

(b) Let z = y 1− n so that: dz dx

 

)

 n dy dx




So that: dz dx

=

(1

n )

(

Q

-

Py

1

n

=

(1

n )

(

Q

-

Pz

)

)

=

Q

1

-

P

1 z

That is: dz dx



P z

1



Q

1

Which can be solved using the integrating factor method.


Learning outcomes




Recognize the order of a differential equation



Appreciate that a differential equation of order n can be derived from a function containing n arbitrary constants



Solve certain first-order differential equations by direct integration



Solve certain first-order differential equations by separating the variables



Solve certain first-order homogeneous differential equations by an appropriate substitution



Solve certain first-order differential equations by using an integrating factor



Solve Bernoulli ’ s equation.


25 First-order differential equations PowerPoint

PROGRAMME 25

FIRST-ORDER

DIFFERENTIAL

EQUATIONS

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FIRST-ORDER

DIFFERENTIAL

EQUATIONS

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