<< Chapter < Page Chapter >> Page >
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] .

d 2 x d t 2 = - c x

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

A x ( t ) = f ( t )

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

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

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

d x d t + r x = 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 1 C q . The voltage across the resistor is R d q d t . Finally, the voltage across the inductor is L d 2 q d t 2 . Therefore, by Kirchoff's voltage law, it follows that

L d 2 q d t 2 + R d q d t + 1 C q = V .
A series RLC circuit.

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

Need help solving this problem (2/7)^-2
Simone Reply
what is the coefficient of -4×
Mehri Reply
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
I'm 13 and I understand it great
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
is it a question of log
I rally confuse this number And equations too I need exactly help
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
greetings from Iran
salut. from Algeria
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
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?