# 3.6 Linear constant coefficient differential equations

 Page 1 / 1
This module describes linear constant coefficient differential equations

## Introduction: ordinary differential equations

In our study of signals and systems, it will often be useful to describe systems using equations involving the rate of change in some quantity. Such equations are called differential equations. For instance, you may remember from a past physics course that an object experiences simple harmonic motion when it has an acceleration that is proportional to the magnitude of its displacement and opposite in direction. Thus, this system is described as the differential equation shown in [link] .

$\frac{{d}^{2}x}{d{t}^{2}}=-cx$

Because the differential equation in [link] has only one independent variable and only has derivatives with respect to that variable, it is called an ordinary differential equation. There are more complicated differential equations, such as the Schrodinger equation, which involve derivatives with respect to multiple independent variables. These are called partial differential equations, but they are not within the scope of this module.

Given a sufficiently descriptive set of initial conditions or boundary conditions, if there is a solution to the differential equation, that solution is unique and describes the behavior of the system. Of course, the results are only accurate to the degree that the model mirrors reality.

## Linear constant coefficient ordinary differential equations

An important subclass of ordinary differential equations is the set of linear constant coefficient ordinary differential equations. These equations are of the form

$Ax\left(t\right)=f\left(t\right)$

where $A$ is a differential operator of the form given in [link] .

$A={a}_{n}\frac{{d}^{n}}{d{t}^{n}}+{a}_{n-1}\frac{{d}^{n-1}}{d{t}^{n-1}}+...+{a}_{1}\frac{d}{dt}+{a}_{0}$

Note that operators of this type satisfy the linearity conditions, and ${a}_{1},...,{a}_{n}$ are real constants. Furthermore, Equation [link] with these operators has derivatives with respect to only one variable, making it an ordinary differential equation.

A similar concept for a discrete time setting, difference equations, is discussed in the chapter on time domain analysis of discrete time systems. There are many parallels between the discussion of linear constant coefficient ordinary differential equations and linear constant coefficient differece equations.

## Applications of differential equations

Consider the decay model in which a quantity of an unstable isotope decreases at a rate proportional to the quanity of unstable isotope remaining. Thus, the decay of the isotope is modeled by the first order linear constant coefficient differential equation

$\frac{dx}{dt}+rx=0$

where $r$ is some real rate.

Now consider the series RLC circuit shown in [link] . This system can be modeled using differential equations. We can use the voltage equations for each circuit element and Kirchoff's voltage law to write a second order linear constant coefficient differential equation describing the charge on the capacitor.

The voltage across the battery is simply $V$ . The voltage across the capacitor is $\frac{1}{C}q$ . The voltage across the resistor is $R\frac{dq}{dt}$ . Finally, the voltage across the inductor is $L\frac{{d}^{2}q}{d{t}^{2}}$ . Therefore, by Kirchoff's voltage law, it follows that

$L\frac{{d}^{2}q}{d{t}^{2}}+R\frac{dq}{dt}+\frac{1}{C}q=V.$

The section Solving Linear Constant Coefficient Differential Equations will describe in depth how solutions to differential equations like those in the examples may be obtained.

## Linear constant coefficient oridinary differential equations summary

Differential equations are an important mathematical tool for modeling continuous time systems. An important subclass of these is the class of linear constant coefficient ordinary differential equations. Linear constant coefficient ordinary differential equations are often particularly easy to solve as will be described in the module on solutions to linear constant coefficient ordinary differential equations and are useful in describing a wide range of situations that arise in electrical engineering and in other fields.

#### Questions & Answers

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
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
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!