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y ( x ) = c 1 e −2 x + c 2 e −7 x .

Single repeated real root

Things are a little more complicated if the characteristic equation has a repeated real root, λ . In this case, we know e λ x is a solution to [link] , but it is only one solution and we need two linearly independent solutions to determine the general solution. We might be tempted to try a function of the form k e λ x , where k is some constant, but it would not be linearly independent of e λ x . Therefore, let’s try x e λ x as the second solution. First, note that by the quadratic formula,

λ = b ± b 2 4 a c 2 a .

But, λ is a repeated root, so b 2 4 a c = 0 and λ = b 2 a . Thus, if y = x e λ x , we have

y = e λ x + λ x e λ x and y = 2 λ e λ x + λ 2 x e λ x .

Substituting these expressions into [link] , we see that

a y + b y + c y = a ( 2 λ e λ x + λ 2 x e λ x ) + b ( e λ x + λ x e λ x ) + c x e λ x = x e λ x ( a λ 2 + b λ + c ) + e λ x ( 2 a λ + b ) = x e λ x ( 0 ) + e λ x ( 2 a ( b 2 a ) + b ) = 0 + e λ x ( 0 ) = 0 .

This shows that x e λ x is a solution to [link] . Since e λ x and x e λ x are linearly independent, when the characteristic equation has a repeated root λ , the general solution to [link] is given by

y ( x ) = c 1 e λ x + c 2 x e λ x ,

where c 1 and c 2 are constants.

For example, the differential equation y + 12 y + 36 y = 0 has the associated characteristic equation λ 2 + 12 λ + 36 = 0 . This factors into ( λ + 6 ) 2 = 0 , which has a repeated root λ = −6 . Therefore, the general solution to this differential equation is

y ( x ) = c 1 e −6 x + c 2 x e −6 x .

Complex conjugate roots

The third case we must consider is when b 2 4 a c < 0 . In this case, when we apply the quadratic formula, we are taking the square root of a negative number. We must use the imaginary number i = −1 to find the roots, which take the form λ 1 = α + β i and λ 2 = α β i . The complex number α + β i is called the conjugate of α β i . Thus, we see that when b 2 4 a c < 0 , the roots of our characteristic equation are always complex conjugates .

This creates a little bit of a problem for us. If we follow the same process we used for distinct real roots—using the roots of the characteristic equation as the coefficients in the exponents of exponential functions—we get the functions e ( α + β i ) x and e ( α β i ) x as our solutions. However, there are problems with this approach. First, these functions take on complex (imaginary) values, and a complete discussion of such functions is beyond the scope of this text. Second, even if we were comfortable with complex-value functions, in this course we do not address the idea of a derivative for such functions. So, if possible, we’d like to find two linearly independent real-value solutions to the differential equation. For purposes of this development, we are going to manipulate and differentiate the functions e ( α + β i ) x and e ( α β i ) x as if they were real-value functions. For these particular functions, this approach is valid mathematically, but be aware that there are other instances when complex-value functions do not follow the same rules as real-value functions. Those of you interested in a more in-depth discussion of complex-value functions should consult a complex analysis text.

Based on the roots α ± β i of the characteristic equation, the functions e ( α + β i ) x and e ( α β i ) x are linearly independent solutions to the differential equation. and the general solution is given by

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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