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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

The auxiliary equation

Summary

Inhomogeneous equations

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:

Programme 25: Second-order differential equations

Introduction

The differential equation:

can be re-written to read:

that is:

Programme 25: Second-order differential equations

Introduction

The differential equation can again be re-written as:

where:

Programme 25: Second-order differential equations

Introduction

The differential equation:

has solution:

This means that:

That is:

Programme 25: Second-order differential equations

Introduction

The differential equation:

has solution:

where:

Programme 25: Second-order differential equations

Introduction

Homogeneous equations

The auxiliary equation

Summary

Inhomogeneous equations

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:

Programme 25: Second-order differential equations

Introduction

Homogeneous equations

The auxiliary equation

Summary

Inhomogeneous equations

Programme 25: Second-order differential equations

The auxiliary equation

Real and different roots

Real and equal roots

Complex roots

Programme 25: Second-order differential equations

The auxiliary equation

Real and different roots

If the auxiliary equation:

with solution:

where:

then the solution to:

Programme 25: Second-order differential equations

The auxiliary equation

Real and equal roots

If the auxiliary equation:

with solution:

where:

then the solution to:

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:

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:

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::

Programme 25: Second-order differential equations

Introduction

Homogeneous equations

The auxiliary equation

Summary

Inhomogeneous equations

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

Programme 25: Second-order differential equations

Introduction

Homogeneous equations

The auxiliary equation

Summary

Inhomogeneous equations

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.

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:

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:

Programme 25: Second-order differential equations

Inhomogeneous equations

Complete solution

(c) The complete solution to:

consists of:

complementary function + particular integral

That is:

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:

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

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