<< Chapter < Page Chapter >> Page >

Examples of first-order nonlinear differential equations include

( y ) 4 ( y ) 3 = ( 3 x 2 ) ( y + 4 ) 4 y + 3 y 3 = 4 x 5 ( y ) 2 = sin y + cos x .

These equations are nonlinear because of terms like ( y ) 4 , y 3 , etc. Due to these terms, it is impossible to put these equations into the same form as [link] .

Standard form

Consider the differential equation

( 3 x 2 4 ) y + ( x 3 ) y = sin x .

Our main goal in this section is to derive a solution method for equations of this form. It is useful to have the coefficient of y be equal to 1 . To make this happen, we divide both sides by 3 x 2 4 .

y + ( x 3 3 x 2 4 ) y = sin x 3 x 2 4

This is called the standard form    of the differential equation. We will use it later when finding the solution to a general first-order linear differential equation. Returning to [link] , we can divide both sides of the equation by a ( x ) . This leads to the equation

y + b ( x ) a ( x ) y = c ( x ) a ( x ) .

Now define p ( x ) = b ( x ) a ( x ) and q ( x ) = c ( x ) a ( x ) . Then [link] becomes

y + p ( x ) y = q ( x ) .

We can write any first-order linear differential equation in this form, and this is referred to as the standard form for a first-order linear differential equation.

Writing first-order linear equations in standard form

Put each of the following first-order linear differential equations into standard form. Identify p ( x ) and q ( x ) for each equation.

  1. y = 3 x 4 y
  2. 3 x y 4 y 3 = 2 (here x > 0 )
  3. y = 3 y 4 x 2 + 5
  1. Add 4 y to both sides:
    y + 4 y = 3 x .

    In this equation, p ( x ) = 4 and q ( x ) = 3 x .
  2. Multiply both sides by 4 y 3 , then subtract 8 y from each side:
    3 x y 4 y 3 = 2 3 x y = 2 ( 4 y 3 ) 3 x y = 8 y 6 3 x y 8 y = −6 .

    Finally, divide both sides by 3 x to make the coefficient of y equal to 1 :
    y 8 3 x y = 2 3 x .
    This is allowable because in the original statement of this problem we assumed that x > 0 . (If x = 0 then the original equation becomes 0 = 2 , which is clearly a false statement.)
    In this equation, p ( x ) = 8 3 x and q ( x ) = 2 3 x .
  3. Subtract y from each side and add 4 x 2 5 :
    3 y y = 4 x 2 5 .

    Next divide both sides by 3 :
    y 1 3 y = 4 3 x 2 5 3 .

    In this equation, p ( x ) = 1 3 and q ( x ) = 4 3 x 2 5 3 .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Put the equation ( x + 3 ) y 2 x 3 y 4 = 5 into standard form and identify p ( x ) and q ( x ) .

y + 15 x + 3 y = 10 x 20 x + 3 ; p ( x ) = 15 x + 3 and q ( x ) = 10 x 20 x + 3

Got questions? Get instant answers now!

Integrating factors

We now develop a solution technique for any first-order linear differential equation. We start with the standard form of a first-order linear differential equation:

y + p ( x ) y = q ( x ) .

The first term on the left-hand side of [link] is the derivative of the unknown function, and the second term is the product of a known function with the unknown function. This is somewhat reminiscent of the power rule from the Differentiation Rules section. If we multiply [link] by a yet-to-be-determined function μ ( x ) , then the equation becomes

μ ( x ) y + μ ( x ) p ( x ) y = μ ( x ) q ( x ) .

The left-hand side [link] can be matched perfectly to the product rule:

d d x [ f ( x ) g ( x ) ] = f ( x ) g ( x ) + f ( x ) g ( x ) .

Matching term by term gives y = f ( x ) , g ( x ) = μ ( x ) , and g ( x ) = μ ( x ) p ( x ) . Taking the derivative of g ( x ) = μ ( x ) and setting it equal to the right-hand side of g ( x ) = μ ( x ) p ( x ) leads to

μ ( x ) = μ ( x ) p ( x ) .

This is a first-order, separable differential equation for μ ( x ) . We know p ( x ) because it appears in the differential equation we are solving. Separating variables and integrating yields

μ ( x ) μ ( x ) = p ( x ) μ ( x ) μ ( x ) d x = p ( x ) d x ln | μ ( x ) | = p ( x ) d x + C e ln | μ ( x ) | = e p ( x ) d x + C | μ ( x ) | = C 1 e p ( x ) d x μ ( x ) = C 2 e p ( x ) d x .

Questions & Answers

what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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
How can I make nanorobot?
Lily
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
how can I make nanorobot?
Lily
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
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
Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
Abdul Reply
You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?
Abdul
Practice Key Terms 3

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 2' conversation and receive update notifications?

Ask