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General solution to a homogeneous equation

If y 1 ( x ) and y 2 ( x ) are linearly independent solutions to a second-order, linear, homogeneous differential equation, then the general solution is given by

y ( x ) = c 1 y 1 ( x ) + c 2 y 2 ( x ) ,

where c 1 and c 2 are constants.

When we say a family of functions is the general solution to a differential equation , we mean that (1) every expression of that form is a solution and (2) every solution to the differential equation can be written in that form, which makes this theorem extremely powerful. If we can find two linearly independent solutions to a differential equation, we have, effectively, found all solutions to the differential equation—quite a remarkable statement. The proof of this theorem is beyond the scope of this text.

Writing the general solution

If y 1 ( t ) = e 3 t and y 2 ( t ) = e −3 t are solutions to y 9 y = 0 , what is the general solution?

Note that y 1 and y 2 are not constant multiples of one another, so they are linearly independent. Then, the general solution to the differential equation is y ( t ) = c 1 e 3 t + c 2 e −3 t .

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If y 1 ( x ) = e 3 x and y 2 ( x ) = x e 3 x are solutions to y 6 y + 9 y = 0 , what is the general solution?

y ( x ) = c 1 e 3 x + c 2 x e 3 x

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Second-order equations with constant coefficients

Now that we have a better feel for linear differential equations, we are going to concentrate on solving second-order equations of the form

a y + b y + c y = 0 ,

where a , b , and c are constants.

Since all the coefficients are constants, the solutions are probably going to be functions with derivatives that are constant multiples of themselves. We need all the terms to cancel out, and if taking a derivative introduces a term that is not a constant multiple of the original function, it is difficult to see how that term cancels out. Exponential functions have derivatives that are constant multiples of the original function, so let’s see what happens when we try a solution of the form y ( x ) = e λ x , where λ (the lowercase Greek letter lambda) is some constant.

If y ( x ) = e λ x , then y ( x ) = λ e λ x and y = λ 2 e λ x . Substituting these expressions into [link] , we get

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

Since e λ x is never zero, this expression can be equal to zero for all x only if

a λ 2 + b λ + c = 0 .

We call this the characteristic equation of the differential equation.


The characteristic equation    of the differential equation a y + b y + c y = 0 is a λ 2 + b λ + c = 0 .

The characteristic equation is very important in finding solutions to differential equations of this form. We can solve the characteristic equation either by factoring or by using the quadratic formula

λ = b ± b 2 4 a c 2 a .

This gives three cases. The characteristic equation has (1) distinct real roots; (2) a single, repeated real root; or (3) complex conjugate roots. We consider each of these cases separately.

Distinct real roots

If the characteristic equation has distinct real roots λ 1 and λ 2 , then e λ 1 x and e λ 2 x are linearly independent solutions to [link] , and the general solution is given by

y ( x ) = c 1 e λ 1 x + c 2 e λ 2 x ,

where c 1 and c 2 are constants.

For example, the differential equation y + 9 y + 14 y = 0 has the associated characteristic equation λ 2 + 9 λ + 14 = 0 . This factors into ( λ + 2 ) ( λ + 7 ) = 0 , which has roots λ 1 = −2 and λ 2 = −7 . Therefore, the general solution to this differential equation is

Questions & Answers

what is the stm
Brian Reply
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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
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 .?
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
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
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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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