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Visit this website that discusses second-order differential equations.

Classify each of the following equations as linear or nonlinear. If the equation is linear, determine further whether it is homogeneous or nonhomogeneous.

  1. ( y ) 2 y + 8 x 3 y = 0
  2. ( sin t ) y + cos t 3 t y = 0
  1. Nonlinear
  2. Linear, nonhomogeneous
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Later in this section, we will see some techniques for solving specific types of differential equations. Before we get to that, however, let’s get a feel for how solutions to linear differential equations behave. In many cases, solving differential equations depends on making educated guesses about what the solution might look like. Knowing how various types of solutions behave will be helpful.

Verifying a solution

Consider the linear, homogeneous differential equation

x 2 y x y 3 y = 0 .

Looking at this equation, notice that the coefficient functions are polynomials, with higher powers of x associated with higher-order derivatives of y . Show that y = x 3 is a solution to this differential equation.

Let y = x 3 . Then y = 3 x 2 and y = 6 x . Substituting into the differential equation, we see that

x 2 y x y 3 y = x 2 ( 6 x ) x ( 3 x 2 ) 3 ( x 3 ) = 6 x 3 3 x 3 3 x 3 = 0 .
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Show that y = 2 x 2 is a solution to the differential equation

1 2 x 2 y x y + y = 0 .
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Although simply finding any solution to a differential equation is important, mathematicians and engineers often want to go beyond finding one solution to a differential equation to finding all solutions to a differential equation. In other words, we want to find a general solution . Just as with first-order differential equations, a general solution (or family of solutions) gives the entire set of solutions to a differential equation. An important difference between first-order and second-order equations is that, with second-order equations, we typically need to find two different solutions to the equation to find the general solution. If we find two solutions, then any linear combination of these solutions is also a solution. We state this fact as the following theorem.

Superposition principle

If y 1 ( x ) and y 2 ( x ) are solutions to a linear homogeneous differential equation, then the function

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

where c 1 and c 2 are constants, is also a solution.

The proof of this superposition principle theorem is left as an exercise.

Verifying the superposition principle

Consider the differential equation

y 4 y 5 y = 0 .

Given that e x and e 5 x are solutions to this differential equation, show that 4 e x + e 5 x is a solution.

We have

y ( x ) = 4 e x + e 5 x , so y ( x ) = −4 e x + 5 e 5 x and y ( x ) = 4 e x + 25 e 5 x .


y 4 y 5 y = ( 4 e x + 25 e 5 x ) 4 ( −4 e x + 5 e 5 x ) 5 ( 4 e x + e 5 x ) = 4 e x + 25 e 5 x + 16 e x 20 e 5 x 20 e x 5 e 5 x = 0 .

Thus, y ( x ) = 4 e x + e 5 x is a solution.

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Consider the differential equation

y + 5 y + 6 y = 0 .

Given that e −2 x and e −3 x are solutions to this differential equation, show that 3 e −2 x + 6 e −3 x is a solution.

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Unfortunately, to find the general solution to a second-order differential equation, it is not enough to find any two solutions and then combine them. Consider the differential equation

x + 7 x + 12 x = 0 .

Both e −3 t and 2 e −3 t are solutions (check this). However, x ( t ) = c 1 e −3 t + c 2 ( 2 e −3 t ) is not the general solution. This expression does not account for all solutions to the differential equation. In particular, it fails to account for the function e −4 t , which is also a solution to the differential equation.

Questions & Answers

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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 .?
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Stoney Reply
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Adin Reply
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Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
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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
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.
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
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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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