Higher Order Linear Equations: Introduction and Basic Results

Let us consider the equation

displaymath28,

and its associated homogeneous equation

displaymath29

The following basic results hold:

(1)
Superposition principle
Let tex2html_wrap_inline42 be solutions of the equation (H). Then, the function

displaymath30

is also solution of the equation (H). This solution is called a linear combination of the functions tex2html_wrap_inline48;

(2)
The general solution of the equation (H) is given by

displaymath31

where tex2html_wrap_inline52 are arbitrary constants and tex2html_wrap_inline54 are n solutions of the equation (H) such that,

displaymath32

In this case, we will say that tex2html_wrap_inline54 are linearly independent. The function tex2html_wrap_inline62 is called the Wronskian of tex2html_wrap_inline64 . We have

displaymath33

Therefore, tex2html_wrap_inline66 , for some tex2html_wrap_inline68, if and only if, tex2html_wrap_inline70 for every x;

(3)
The general solution of the equation (NH) is given by

displaymath34

where tex2html_wrap_inline52 are arbitrary constants, tex2html_wrap_inline54 are linearly independents solutions of the associated homogeneous equation (H), and tex2html_wrap_inline82 is a particular solution of (NH).

[Differential Equations] [First Order D.E.] [Second Order D.E.]
[Geometry] [Algebra] [Trigonometry ]
[Calculus] [Complex Variables] [Matrix Algebra]

S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Author: Mohamed Amine Khamsi

Copyright 1999-2017 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour