<< Chapter < Page Chapter >> Page >
This module contains information on solving linear constant coefficient differential equations.

Introduction

The approach to solving linear constant coefficient ordinary differential equations is to find the general form of all possible solutions to the equation and then apply a number of conditions to find the appropriate solution. The two main types of problems are initial value problems, which involve constraints on the solution and its derivatives at a single point, and boundary value problems, which involve constraints on the solution or its derivatives at several points.

The number of initial conditions needed for an N th order differential equation, which is the order of the highest order derivative, is N , and a unique solution is always guaranteed if these are supplied. Boundary value problems can be slightly more complicated and will not necessarily have a unique solution or even a solution at all for a given set of conditions. Thus, this module will focus exclusively on initial value problems.

Solving linear constant coefficient ordinary differential equations

Consider some linear constant coefficient ordinary differential equation given by A x ( t ) = f ( t ) , where A is a differential operator of the form

A = a n d n d t n + a n - 1 d n - 1 d t n - 1 + . . . + a 1 d d t + a 0 .

Let x h ( t ) and x p ( t ) be two functions such that A x h ( t ) = 0 and A x p ( t ) = f ( t ) . By the linearity of A , note that A ( x h ( t ) + x p ( t ) ) = 0 + f ( t ) = f ( t ) . Thus, the form of the general solution x g ( t ) to any linear constant coefficient ordinary differential equation is the sum of a homogeneous solution x h ( t ) to the equation A x = 0 and a particular solution x p ( t ) that is specific to the forcing function f ( t ) .

We wish to determine the forms of the homogeneous and nonhomogeneous solutions in full generality in order to avoid incorrectly restricting the form of the solution before applying any conditions. Otherwise, a valid set of initial or boundary conditions might appear to have no corresponding solution trajectory. The following discussion shows how to accomplish this for linear constant coefficient ordinary differential equations.

Finding the homogeneous solution

In order to find the homogeneous solution to A x ( t ) = f ( t ) , consider the differential equation A x ( t ) = 0 . We know that the solutions have the form c e λ t for some complex constants c , λ . Since A c e λ t = 0 for a solution, it follows that

a n d n d t n + a n - 1 d n - 1 d t n - 1 + . . . + a 1 d d t + a 0 e λ t = 0 ,

so it also follows that

a n λ n + a n - 1 λ n - 1 . . . + a 1 λ + a 0 = 0 .

Therefore, the parameters of the solution exponents are the roots of the above polynomial, called the characteristic polynomial.

For equations of order two or more, there will be several roots. If all of the roots are distinct, then the the general form of the homogeneous solution is simply

x h ( t ) = c 1 e λ 1 t + . . . + c n e λ n t .

If a root has multiplicity that is greater than one, the repeated solutions must be multiplied by each powers of t from 0 to one less than the root multiplicity (in order to ensure linearly independent solutions). For instance, if λ 1 had multiplicity 2 and λ 2 had multiplicity 3, the homogeneous solution would be

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signals and systems' conversation and receive update notifications?

Ask