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# Linear with Constant Coefficient

A guide to linear equations of second and higher degrees
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## Definition

The Equations in this section are of the form:

where is a function of x but all of the 's are Constants.

These equations are of the utmost importance in the study of vibrations of all kinds(Mechanics; Acoustics and Electrical). The methods given are chiefly due to Euler and D'Alembert.

## Equation Of The First Order

If and equation (2) becomes

Therefore

Integrating

Therefore

Let the Constant equal Thus

Therefore is the general solution for the first order differential equation .

## Equations Of The Second Order

If and

Equation (2) can now be written as:

The solution to equation (3) suggests that where m is some constant may satisfy equation (4). With this value for equation (4) reduces to:

Thus if is a root of:

is a solution of equation (4) whatever the value of A

Let the roots of equation (5) be and .

If the roots are unequal we will have two solutions to equation (4) namely and
Then the general solution will be

If the roots are equal we will also have two solutions to equation (4) namely and
Then the general solution will be

Equation (5) is called the "Auxiliary Equation"
As an example, to solve
Therefore

This equation is satisfied by or

The General Solution is therefore given by:

## Modifications When The Auxiliary Equation Has Imaginary Or Complex Roots

When the auxiliary equation (5) has roots of the form and where

it is best to modify the solution

so that it does not contain imaginary quantities.. To do this use the following trigonometrical identities:

Thus equation (6) becomes

Writing for and for

and are arbitrary constants as were and . It might look as if F must be imaginary but this is not necessarily so . Thus if and then and .

Example:

##### Example - Second degree equation
Problem

Workings
From this the auxiliary equation is:

and the roots are

The solution can be written as;-

Solution
or in a more useful form:

Or

Where

So that

## The Extension To Orders Higher Than The Second

The methods discussed in this section apply to equation (2) whatever the value of n provided that
Example:

##### Example - Third degree equation
Problem

Workings
The Auxiliary Equation is:

Solution
Thus m = 1, 2, or 3 Therefore

## The Complementary Function And The Particular Integral

So far we have only dealt with examples where the of equation (2) has been zero. It will now be shown that the relation between the solution of the equation when is not zero and the solution of a simpler equation derived from it by replacing by zero.

Consider the equation:

By inspection it can be seen that y = x is one solution. Such a solution containing no arbitrary constants is called a Particular Integral

Now substitute in the equation which becomes:

From this it can be shown that :

Note
The general solution of a linear differential equation with constant coefficients is the sum of a Particular Integral and the Complementary Function, the latter being the solution of the equation obtained by substituting zero for the function of x occurring.

The terms containing the arbitrary constants are called the Complementary Function

This can be expressed in a general form.

If is a particular integral of :

So that:

Putting in equation (7) and subtracting equation (8) gives:

If the solution to this equation is contains n arbitrary constants then the general solution to equation (7) is :

and is called the Complementary Function.