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# PROGRAMME 25 - PowerPoint PPT Presentation

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

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SECOND-ORDER DIFFERENTIAL EQUATIONS

Introduction

Homogeneous equations

The auxiliary equation

Summary

Inhomogeneous equations

Introduction

Homogeneous equations

The auxiliary equation

Summary

Inhomogeneous equations

Introduction

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

are solutions to the quadratic equation:

with the properties:

Introduction

The differential equation:

that is:

Introduction

The differential equation can again be re-written as:

where:

Introduction

The differential equation:

has solution:

This means that:

That is:

Introduction

The differential equation:

has solution:

where:

Introduction

Homogeneous equations

The auxiliary equation

Summary

Inhomogeneous 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:

Introduction

Homogeneous equations

The auxiliary equation

Summary

Inhomogeneous equations

The auxiliary equation

Real and different roots

Real and equal roots

Complex roots

The auxiliary equation

Real and different roots

If the auxiliary equation:

with solution:

where:

then the solution to:

The auxiliary equation

Real and equal roots

If the auxiliary equation:

with solution:

where:

then the solution to:

The auxiliary equation

Complex roots

If the auxiliary equation:

with solution:

where:

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

The auxiliary equation

Complex roots

Complex roots to the auxiliary equation:

means that the solution of the differential equation:

is of the form:

The auxiliary equation

Complex roots

Since:

then:

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

Introduction

Homogeneous equations

The auxiliary equation

Summary

Inhomogeneous equations

Summary

Differential equations of the form:

Auxiliary equation:

Roots real and different: Solution

Roots real and the same: Solution

Roots complex (  j): Solution

Introduction

Homogeneous equations

The auxiliary equation

Summary

Inhomogeneous 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.

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:

Inhomogeneous equations

Particular integral

(b) Particular integral

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

gives:

yielding:

so that:

Inhomogeneous equations

Complete solution

(c) The complete solution to:

consists of:

complementary function + particular integral

That is:

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:

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