# 7.1 Second-order linear equations  (Page 7/15)

 Page 7 / 15

Find the general solution to the following differential equations:

1. $y\text{″}-2{y}^{\prime }+10y=0$
2. $y\text{″}+14{y}^{\prime }+49y=0$
1. $y\left(x\right)={e}^{x}\left({c}_{1}\text{cos}\phantom{\rule{0.1em}{0ex}}3x+{c}_{2}\text{sin}\phantom{\rule{0.1em}{0ex}}3x\right)$
2. $y\left(x\right)={c}_{1}{e}^{-7x}+{c}_{2}x{e}^{-7x}$

## Initial-value problems and boundary-value problems

So far, we have been finding general solutions to differential equations. However, differential equations are often used to describe physical systems, and the person studying that physical system usually knows something about the state of that system at one or more points in time. For example, if a constant-coefficient differential equation is representing how far a motorcycle shock absorber is compressed, we might know that the rider is sitting still on his motorcycle at the start of a race, time $t={t}_{0}.$ This means the system is at equilibrium, so $y\left({t}_{0}\right)=0,$ and the compression of the shock absorber is not changing, so ${y}^{\prime }\left({t}_{0}\right)=0.$ With these two initial conditions and the general solution to the differential equation, we can find the specific solution to the differential equation that satisfies both initial conditions. This process is known as solving an initial-value problem . (Recall that we discussed initial-value problems in Introduction to Differential Equations .) Note that second-order equations have two arbitrary constants in the general solution, and therefore we require two initial conditions to find the solution to the initial-value problem.

Sometimes we know the condition of the system at two different times. For example, we might know $y\left({t}_{0}\right)={y}_{0}$ and $y\left({t}_{1}\right)={y}_{1}.$ These conditions are called boundary conditions    , and finding the solution to the differential equation that satisfies the boundary conditions is called solving a boundary-value problem    .

Mathematicians, scientists, and engineers are interested in understanding the conditions under which an initial-value problem or a boundary-value problem has a unique solution. Although a complete treatment of this topic is beyond the scope of this text, it is useful to know that, within the context of constant-coefficient, second-order equations, initial-value problems are guaranteed to have a unique solution as long as two initial conditions are provided. Boundary-value problems, however, are not as well behaved. Even when two boundary conditions are known, we may encounter boundary-value problems with unique solutions, many solutions, or no solution at all.

## Solving an initial-value problem

Solve the following initial-value problem: $y\text{″}+3{y}^{\prime }-4y=0,$ $y\left(0\right)=1,$ ${y}^{\prime }\left(0\right)=-9.$

We already solved this differential equation in [link] a. and found the general solution to be

$y\left(x\right)={c}_{1}{e}^{-4x}+{c}_{2}{e}^{x}.$

Then

${y}^{\prime }\left(x\right)=-4{c}_{1}{e}^{-4x}+{c}_{2}{e}^{x}.$

When $x=0,$ we have $y\left(0\right)={c}_{1}+{c}_{2}$ and ${y}^{\prime }\left(0\right)=-4{c}_{1}+{c}_{2}.$ Applying the initial conditions, we have

$\begin{array}{ccc}\hfill {c}_{1}+{c}_{2}& =\hfill & 1\hfill \\ \hfill -4{c}_{1}+{c}_{2}& =\hfill & -9.\hfill \end{array}$

Then ${c}_{1}=1-{c}_{2}.$ Substituting this expression into the second equation, we see that

$\begin{array}{ccc}\hfill -4\left(1-{c}_{2}\right)+{c}_{2}& =\hfill & -9\hfill \\ \hfill -4+4{c}_{2}+{c}_{2}& =\hfill & -9\hfill \\ \hfill 5{c}_{2}& =\hfill & -5\hfill \\ \hfill {c}_{2}& =\hfill & -1.\hfill \end{array}$

So, ${c}_{1}=2$ and the solution to the initial-value problem is

$y\left(x\right)=2{e}^{-4x}-{e}^{x}.$

Solve the initial-value problem $y\text{″}-3{y}^{\prime }-10y=0,$ $y\left(0\right)=0,$ ${y}^{\prime }\left(0\right)=7.$

$y\left(x\right)=\text{−}{e}^{-2x}+{e}^{5x}$

## Solving an initial-value problem and graphing the solution

Solve the following initial-value problem and graph the solution:

$y\text{″}+6{y}^{\prime }+13y=0,\phantom{\rule{0.2em}{0ex}}y\left(0\right)=0,\phantom{\rule{0.2em}{0ex}}{y}^{\prime }\left(0\right)=2$

We already solved this differential equation in [link] b. and found the general solution to be

$y\left(x\right)={e}^{-3x}\left({c}_{1}\text{cos}\phantom{\rule{0.1em}{0ex}}2x+{c}_{2}\text{sin}\phantom{\rule{0.1em}{0ex}}2x\right).$

Then

${y}^{\prime }\left(x\right)={e}^{-3x}\left(-2{c}_{1}\text{sin}\phantom{\rule{0.1em}{0ex}}2x+2{c}_{2}\text{cos}\phantom{\rule{0.1em}{0ex}}2x\right)-3{e}^{-3x}\left({c}_{1}\text{cos}\phantom{\rule{0.1em}{0ex}}2x+{c}_{2}\text{sin}\phantom{\rule{0.1em}{0ex}}2x\right).$

When $x=0,$ we have $y\left(0\right)={c}_{1}$ and ${y}^{\prime }\left(0\right)=2{c}_{2}-3{c}_{1}.$ Applying the initial conditions, we obtain

$\begin{array}{ccc}\hfill {c}_{1}& =\hfill & 0\hfill \\ \hfill -3{c}_{1}+2{c}_{2}& =\hfill & 2.\hfill \end{array}$

Therefore, ${c}_{1}=0,$ ${c}_{2}=1,$ and the solution to the initial value problem is shown in the following graph.

$y={e}^{-3x}\text{sin}\phantom{\rule{0.1em}{0ex}}2x\text{.}$ #### Questions & Answers

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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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
Do you know which machine is used to that process?
s.
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
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!
Jobilize.com Reply

### Read also:

#### 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?    By Rhodes     By Anonymous User 