<< 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

how to create a software using Android phone
Wiseman Reply
how
basra
what is the difference between C and C++.
Yan Reply
what is software
Sami Reply
software is a instructions like programs
Shambhu
what is the difference between C and C++.
Yan
yes, how?
Hayder
what is software engineering
Ahmad
software engineering is a the branch of computer science deals with the design,development, testing and maintenance of software applications.
Hayder
who is best bw software engineering and cyber security
Ahmad
Both software engineering and cybersecurity offer exciting career prospects, but your choice ultimately depends on your interests and skills. If you enjoy problem-solving, programming, and designing software syste
Hayder
what's software processes
Ntege Reply
I haven't started reading yet. by device (hardware) or for improving design Lol? Here. Requirement, Design, Implementation, Verification, Maintenance.
Vernon
I can give you a more valid answer by 5:00 By the way gm.
Vernon
it is all about designing,developing, testing, implementing and maintaining of software systems.
Ehenew
hello assalamualaikum
Sami
My name M Sami I m 2nd year student
Sami
what is the specific IDE for flutter programs?
Mwami Reply
jegudgdtgd my Name my Name is M and I have been talking about iey my papa john's university of washington post I tagged I will be in
Mwaqas Reply
yes
usman
how disign photo
atul Reply
hlo
Navya
hi
Michael
yes
Subhan
Show the necessary steps with description in resource monitoring process (CPU,memory,disk and network)
samuel Reply
What is software engineering
Tafadzwa Reply
Software engineering is a branch of computer science directed to writing programs to develop Softwares that can drive or enable the functionality of some hardwares like phone , automobile and others
kelvin
if any requirement engineer is gathering requirements from client and after getting he/she Analyze them this process is called
Alqa Reply
The following text is encoded in base 64. Ik5ldmVyIHRydXN0IGEgY29tcHV0ZXIgeW91IGNhbid0IHRocm93IG91dCBhIHdpbmRvdyIgLSBTdGV2ZSBXb3puaWFr Decode it, and paste the decoded text here
Julian Reply
what to do you mean
Vincent
hello
ALI
how are you ?
ALI
What is the command to list the contents of a directory in Unix and Unix-like operating systems
George Reply
how can i make my own software free of cost
Faizan Reply
like how
usman
hi
Hayder
The name of the author of our software engineering book is Ian Sommerville.
Doha Reply
what is software
Sampson Reply
the set of intruction given to the computer to perform a task
Noor
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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