Linear with Constant Coefficient
A guide to linear equations of second and higher degrees
Contents
Definition
The Equations in this section are of the form: whereEquations Of The Second Order
IfIf 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"
Then the general solution will be
Then the general solution will be
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 formExample:
Example - Second degree equation
Problem
Workings
From this the auxiliary equation is:
and the roots are
The solution can be written as;-
x}\;+B\;e^{(3-2i)x})
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 thatExample:
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 theNote
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 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.