# 7.1 Second-order linear equations  (Page 5/15)

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$y\left(x\right)={c}_{1}{e}^{-2x}+{c}_{2}{e}^{-7x}.$

## Single repeated real root

Things are a little more complicated if the characteristic equation has a repeated real root, $\lambda \text{.}$ In this case, we know ${e}^{\lambda x}$ 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 $k{e}^{\lambda x},$ where k is some constant, but it would not be linearly independent of ${e}^{\lambda x}.$ Therefore, let’s try $x{e}^{\lambda x}$ as the second solution. First, note that by the quadratic formula,

$\lambda =\frac{\text{−}b±\sqrt{{b}^{2}-4ac}}{2a}.$

But, $\lambda$ is a repeated root, so ${b}^{2}-4ac=0$ and $\lambda =\frac{\text{−}b}{2a}.$ Thus, if $y=x{e}^{\lambda x},$ we have

${y}^{\prime }={e}^{\lambda x}+\lambda x{e}^{\lambda x}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y\text{″}=2\lambda {e}^{\lambda x}+{\lambda }^{2}x{e}^{\lambda x}.$

Substituting these expressions into [link] , we see that

$\begin{array}{cc}\hfill ay\text{″}+b{y}^{\prime }+cy& =a\left(2\lambda {e}^{\lambda x}+{\lambda }^{2}x{e}^{\lambda x}\right)+b\left({e}^{\lambda x}+\lambda x{e}^{\lambda x}\right)+cx{e}^{\lambda x}\hfill \\ & =x{e}^{\lambda x}\left(a{\lambda }^{2}+b\lambda +c\right)+{e}^{\lambda x}\left(2a\lambda +b\right)\hfill \\ & =x{e}^{\lambda x}\left(0\right)+{e}^{\lambda x}\left(2a\left(\frac{\text{−}b}{2a}\right)+b\right)\hfill \\ & =0+{e}^{\lambda x}\left(0\right)\hfill \\ & =0.\hfill \end{array}$

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

$y\left(x\right)={c}_{1}{e}^{\lambda x}+{c}_{2}x{e}^{\lambda x},$

where ${c}_{1}$ and ${c}_{2}$ are constants.

For example, the differential equation $y\text{″}+12{y}^{\prime }+36y=0$ has the associated characteristic equation ${\lambda }^{2}+12\lambda +36=0.$ This factors into ${\left(\lambda +6\right)}^{2}=0,$ which has a repeated root $\lambda =-6.$ Therefore, the general solution to this differential equation is

$y\left(x\right)={c}_{1}{e}^{-6x}+{c}_{2}x{e}^{-6x}.$

## Complex conjugate roots

The third case we must consider is when ${b}^{2}-4ac<0.$ 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 $i=\sqrt{-1}$ to find the roots, which take the form ${\lambda }_{1}=\alpha +\beta i$ and ${\lambda }_{2}=\alpha -\beta i\text{.}$ The complex number $\alpha +\beta i$ is called the conjugate of $\alpha -\beta i\text{.}$ Thus, we see that when ${b}^{2}-4ac<0,$ 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 ${e}^{\left(\alpha +\beta i\right)x}$ and ${e}^{\left(\alpha -\beta i\right)x}$ 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 ${e}^{\left(\alpha +\beta i\right)x}$ and ${e}^{\left(\alpha -\beta i\right)x}$ 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 $\alpha ±\beta i$ of the characteristic equation, the functions ${e}^{\left(\alpha +\beta i\right)x}$ and ${e}^{\left(\alpha -\beta i\right)x}$ are linearly independent solutions to the differential equation. and the general solution is given by

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
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?
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