PROGRAMME 25
Download
1 / 26

PROGRAMME 25 - PowerPoint PPT Presentation


  • 186 Views
  • Uploaded on

PROGRAMME 25. SECOND-ORDER DIFFERENTIAL EQUATIONS. Programme 25: Second-order differential equations. Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations. Programme 25: Second-order differential equations. Introduction Homogeneous equations

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'PROGRAMME 25' - king


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Slide1 l.jpg

PROGRAMME 25

SECOND-ORDER DIFFERENTIAL EQUATIONS


Slide2 l.jpg

Programme 25: Second-order differential equations

Introduction

Homogeneous equations

The auxiliary equation

Summary

Inhomogeneous equations


Slide3 l.jpg

Programme 25: Second-order differential equations

Introduction

Homogeneous equations

The auxiliary equation

Summary

Inhomogeneous equations


Slide4 l.jpg

Programme 25: Second-order differential equations

Introduction

For any three numbers a, b and c, the two numbers:

are solutions to the quadratic equation:

with the properties:


Slide5 l.jpg

Programme 25: Second-order differential equations

Introduction

The differential equation:

can be re-written to read:

that is:


Slide6 l.jpg

Programme 25: Second-order differential equations

Introduction

The differential equation can again be re-written as:

where:


Slide7 l.jpg

Programme 25: Second-order differential equations

Introduction

The differential equation:

has solution:

This means that:

That is:


Slide8 l.jpg

Programme 25: Second-order differential equations

Introduction

The differential equation:

has solution:

where:


Slide9 l.jpg

Programme 25: Second-order differential equations

Introduction

Homogeneous equations

The auxiliary equation

Summary

Inhomogeneous equations


Slide10 l.jpg

Programme 25: Second-order differential equations

Homogeneous equations

The differential equation:

Is a second-order, constant coefficient, linear, homogeneous differential equation. Its solution is found from the solutions to the auxiliary equation:

These are:


Slide11 l.jpg

Programme 25: Second-order differential equations

Introduction

Homogeneous equations

The auxiliary equation

Summary

Inhomogeneous equations


Slide12 l.jpg

Programme 25: Second-order differential equations

The auxiliary equation

Real and different roots

Real and equal roots

Complex roots


Slide13 l.jpg

Programme 25: Second-order differential equations

The auxiliary equation

Real and different roots

If the auxiliary equation:

with solution:

where:

then the solution to:


Slide14 l.jpg

Programme 25: Second-order differential equations

The auxiliary equation

Real and equal roots

If the auxiliary equation:

with solution:

where:

then the solution to:


Slide15 l.jpg

Programme 25: Second-order differential equations

The auxiliary equation

Complex roots

If the auxiliary equation:

with solution:

where:

Then the solutions to the auxiliary equation are complex conjugates. That is:


Slide16 l.jpg

Programme 25: Second-order differential equations

The auxiliary equation

Complex roots

Complex roots to the auxiliary equation:

means that the solution of the differential equation:

is of the form:


Slide17 l.jpg

Programme 25: Second-order differential equations

The auxiliary equation

Complex roots

Since:

then:

The solution to the differential equation whose auxiliary equation has complex roots can be written as::


Slide18 l.jpg

Programme 25: Second-order differential equations

Introduction

Homogeneous equations

The auxiliary equation

Summary

Inhomogeneous equations


Slide19 l.jpg

Programme 25: Second-order differential equations

Summary

Differential equations of the form:

Auxiliary equation:

Roots real and different: Solution

Roots real and the same: Solution

Roots complex (  j): Solution


Slide20 l.jpg

Programme 25: Second-order differential equations

Introduction

Homogeneous equations

The auxiliary equation

Summary

Inhomogeneous equations


Slide21 l.jpg

Programme 25: Second-order differential equations

Inhomogeneous equations

  • The second-order, constant coefficient, linear, inhomogeneous differential

  • equation is an equation of the type:

  • The solution is in two parts y1 + y2:

  • part 1, y1 is the solution to the homogeneous equation and is called the complementary function which is the solution to the homogeneous equation

  • part 2, y2 is called the particular integral.


Slide22 l.jpg

Programme 25: Second-order differential equations

Inhomogeneous equations

Complementary function

  • Example, to solve:

  • Complementary function

  • Auxiliary equation: m2 – 5m + 6 = 0 solution m = 2, 3

  • Complementary function y1 = Ae2x + Be3x where:


Slide23 l.jpg

Programme 25: Second-order differential equations

Inhomogeneous equations

Particular integral

(b) Particular integral

Assume a form for y2 as y2 = Cx2 + Dx + E then substitution in:

gives:

yielding:

so that:


Slide24 l.jpg

Programme 25: Second-order differential equations

Inhomogeneous equations

Complete solution

(c) The complete solution to:

consists of:

complementary function + particular integral

That is:


Slide25 l.jpg

Programme 25: Second-order differential equations

Inhomogeneous equations

Particular integrals

The general form assumed for the particular integral depends upon the form of

the right-hand side of the inhomogeneous equation. The following table can be

used as a guide:


Slide26 l.jpg

Programme 25: Second-order differential equations

Learning outcomes

  • Use the auxiliary equation to solve certain second-order homogeneous equations

  • Use the complementary function and the particular integral to solve certain second-order inhomogeneous equations


ad