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Find the general solution to the following differential equations:

  1. y 2 y + 10 y = 0
  2. y + 14 y + 49 y = 0
  1. y ( x ) = e x ( c 1 cos 3 x + c 2 sin 3 x )
  2. y ( x ) = c 1 e −7 x + c 2 x e −7 x
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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 ( t 0 ) = 0 , and the compression of the shock absorber is not changing, so y ( t 0 ) = 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 ( t 0 ) = y 0 and y ( t 1 ) = 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 + 3 y 4 y = 0 , y ( 0 ) = 1 , y ( 0 ) = −9 .

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

y ( x ) = c 1 e −4 x + c 2 e x .


y ( x ) = −4 c 1 e −4 x + c 2 e x .

When x = 0 , we have y ( 0 ) = c 1 + c 2 and y ( 0 ) = −4 c 1 + c 2 . Applying the initial conditions, we have

c 1 + c 2 = 1 −4 c 1 + c 2 = −9 .

Then c 1 = 1 c 2 . Substituting this expression into the second equation, we see that

−4 ( 1 c 2 ) + c 2 = −9 −4 + 4 c 2 + c 2 = −9 5 c 2 = −5 c 2 = −1 .

So, c 1 = 2 and the solution to the initial-value problem is

y ( x ) = 2 e −4 x e x .
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Solve the initial-value problem y 3 y 10 y = 0 , y ( 0 ) = 0 , y ( 0 ) = 7 .

y ( x ) = e −2 x + e 5 x

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Solving an initial-value problem and graphing the solution

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

y + 6 y + 13 y = 0 , y ( 0 ) = 0 , y ( 0 ) = 2

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

y ( x ) = e −3 x ( c 1 cos 2 x + c 2 sin 2 x ) .


y ( x ) = e −3 x ( −2 c 1 sin 2 x + 2 c 2 cos 2 x ) 3 e −3 x ( c 1 cos 2 x + c 2 sin 2 x ) .

When x = 0 , we have y ( 0 ) = c 1 and y ( 0 ) = 2 c 2 3 c 1 . Applying the initial conditions, we obtain

c 1 = 0 −3 c 1 + 2 c 2 = 2 .

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

y = e −3 x sin 2 x .
This figure is a graph of the function y = e^−3x sin 2x. The x axis is scaled in increments of tenths. The y axis is scaled in increments of even tenths. The curve passes through the origin and has a horizontal asymptote of the positive x axis.
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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
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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.
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Damian Reply
absolutely yes
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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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 ?
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian 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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