<< Chapter < Page Chapter >> Page >

It turns out that to find the general solution to a second-order differential equation, we must find two linearly independent solutions. We define that terminology here.

Definition

A set of functions f 1 ( x ) , f 2 ( x ),…, f n ( x ) is said to be linearly dependent    if there are constants c 1 , c 2 ,… c n , not all zero, such that c 1 f 1 ( x ) + c 2 f 2 ( x ) + + c n f n ( x ) = 0 for all x over the interval of interest. A set of functions that is not linearly dependent is said to be linearly independent    .

In this chapter, we usually test sets of only two functions for linear independence, which allows us to simplify this definition. From a practical perspective, we see that two functions are linearly dependent if either one of them is identically zero or if they are constant multiples of each other.

First we show that if the functions meet the conditions given previously, then they are linearly dependent. If one of the functions is identically zero—say, f 2 ( x ) 0 —then choose c 1 = 0 and c 2 = 1 , and the condition for linear dependence is satisfied. If, on the other hand, neither f 1 ( x ) nor f 2 ( x ) is identically zero, but f 1 ( x ) = C f 2 ( x ) for some constant C , then choose c 1 = 1 C and c 2 = −1 , and again, the condition is satisfied.

Next, we show that if two functions are linearly dependent, then either one is identically zero or they are constant multiples of one another. Assume f 1 ( x ) and f 2 ( x ) are linearly independent. Then, there are constants, c 1 and c 2 , not both zero, such that

c 1 f 1 ( x ) + c 2 f 2 ( x ) = 0

for all x over the interval of interest. Then,

c 1 f 1 ( x ) = c 2 f 2 ( x ) .

Now, since we stated that c 1 and c 2 can’t both be zero, assume c 2 0 . Then, there are two cases: either c 1 = 0 or c 1 0 . If c 1 = 0 , then

0 = c 2 f 2 ( x ) 0 = f 2 ( x ) ,

so one of the functions is identically zero. Now suppose c 1 0 . Then,

f 1 ( x ) = ( c 2 c 1 ) f 2 ( x )

and we see that the functions are constant multiples of one another.

Linear dependence of two functions

Two functions, f 1 ( x ) and f 2 ( x ) , are said to be linearly dependent if either one of them is identically zero or if f 1 ( x ) = C f 2 ( x ) for some constant C and for all x over the interval of interest. Functions that are not linearly dependent are said to be linearly independent .

Testing for linear dependence

Determine whether the following pairs of functions are linearly dependent or linearly independent.

  1. f 1 ( x ) = x 2 , f 2 ( x ) = 5 x 2
  2. f 1 ( x ) = sin x , f 2 ( x ) = cos x
  3. f 1 ( x ) = e 3 x , f 2 ( x ) = e −3 x
  4. f 1 ( x ) = 3 x , f 2 ( x ) = 3 x + 1
  1. f 2 ( x ) = 5 f 1 ( x ) , so the functions are linearly dependent.
  2. There is no constant C such that f 1 ( x ) = C f 2 ( x ) , so the functions are linearly independent.
  3. There is no constant C such that f 1 ( x ) = C f 2 ( x ) , so the functions are linearly independent. Don’t get confused by the fact that the exponents are constant multiples of each other. With two exponential functions, unless the exponents are equal, the functions are linearly independent.
  4. There is no constant C such that f 1 ( x ) = C f 2 ( x ) , so the functions are linearly independent.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Determine whether the following pairs of functions are linearly dependent or linearly independent: f 1 ( x ) = e x , f 2 ( x ) = 3 e 3 x .

Linearly independent

Got questions? Get instant answers now!

If we are able to find two linearly independent solutions to a second-order differential equation, then we can combine them to find the general solution. This result is formally stated in the following theorem.

Questions & Answers

Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply
Practice Key Terms 7

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?

Ask