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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

are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?
Radek Reply
Practice Key Terms 7

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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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