<< Chapter < Page Chapter >> Page >

Use power series to solve y = 2 y , y ( 0 ) = 5 .

y = 5 e 2 x

Got questions? Get instant answers now!

We now consider an example involving a differential equation that we cannot solve using previously discussed methods. This differential equation

y x y = 0

is known as Airy’s equation . It has many applications in mathematical physics, such as modeling the diffraction of light. Here we show how to solve it using power series.

Power series solution of airy’s equation

Use power series to solve

y x y = 0

with the initial conditions y ( 0 ) = a and y ( 0 ) = b .

We look for a solution of the form

y = n = 0 c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + c 4 x 4 + .

Differentiating this function term by term, we obtain

y = c 1 + 2 c 2 x + 3 c 3 x 2 + 4 c 4 x 3 + , y = 2 · 1 c 2 + 3 · 2 c 3 x + 4 · 3 c 4 x 2 + .

If y satisfies the equation y = x y , then

2 · 1 c 2 + 3 · 2 c 3 x + 4 · 3 c 4 x 2 + = x ( c 0 + c 1 x + c 2 x 2 + c 3 x 3 + ) .

Using [link] on the uniqueness of power series representations, we know that coefficients of the same degree must be equal. Therefore,

2 · 1 c 2 = 0 , 3 · 2 c 3 = c 0 , 4 · 3 c 4 = c 1 , 5 · 4 c 5 = c 2 , .

More generally, for n 3 , we have n · ( n 1 ) c n = c n 3 . In fact, all coefficients can be written in terms of c 0 and c 1 . To see this, first note that c 2 = 0 . Then

c 3 = c 0 3 · 2 , c 4 = c 1 4 · 3 .

For c 5 , c 6 , c 7 , we see that

c 5 = c 2 5 · 4 = 0 , c 6 = c 3 6 · 5 = c 0 6 · 5 · 3 · 2 , c 7 = c 4 7 · 6 = c 1 7 · 6 · 4 · 3 .

Therefore, the series solution of the differential equation is given by

y = c 0 + c 1 x + 0 · x 2 + c 0 3 · 2 x 3 + c 1 4 · 3 x 4 + 0 · x 5 + c 0 6 · 5 · 3 · 2 x 6 + c 1 7 · 6 · 4 · 3 x 7 + .

The initial condition y ( 0 ) = a implies c 0 = a . Differentiating this series term by term and using the fact that y ( 0 ) = b , we conclude that c 1 = b . Therefore, the solution of this initial-value problem is

y = a ( 1 + x 3 3 · 2 + x 6 6 · 5 · 3 · 2 + ) + b ( x + x 4 4 · 3 + x 7 7 · 6 · 4 · 3 + ) .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Use power series to solve y + x 2 y = 0 with the initial condition y ( 0 ) = a and y ( 0 ) = b .

y = a ( 1 x 4 3 · 4 + x 8 3 · 4 · 7 · 8 ) + b ( x x 5 4 · 5 + x 9 4 · 5 · 8 · 9 )

Got questions? Get instant answers now!

Evaluating nonelementary integrals

Solving differential equations is one common application of power series. We now turn to a second application. We show how power series can be used to evaluate integrals involving functions whose antiderivatives cannot be expressed using elementary functions.

One integral that arises often in applications in probability theory is e x 2 d x . Unfortunately, the antiderivative of the integrand e x 2 is not an elementary function. By elementary function, we mean a function that can be written using a finite number of algebraic combinations or compositions of exponential, logarithmic, trigonometric, or power functions. We remark that the term “elementary function” is not synonymous with noncomplicated function. For example, the function f ( x ) = x 2 3 x + e x 3 sin ( 5 x + 4 ) is an elementary function, although not a particularly simple-looking function. Any integral of the form f ( x ) d x where the antiderivative of f cannot be written as an elementary function is considered a nonelementary integral    .

Nonelementary integrals cannot be evaluated using the basic integration techniques discussed earlier. One way to evaluate such integrals is by expressing the integrand as a power series and integrating term by term. We demonstrate this technique by considering e x 2 d x .

Using taylor series to evaluate a definite integral

  1. Express e x 2 d x as an infinite series.
  2. Evaluate 0 1 e x 2 d x to within an error of 0.01 .
  1. The Maclaurin series for e x 2 is given by
    e x 2 = n = 0 ( x 2 ) n n ! = 1 x 2 + x 4 2 ! x 6 3 ! + + ( −1 ) n x 2 n n ! + = n = 0 ( −1 ) n x 2 n n ! .

    Therefore,
    e x 2 d x = ( 1 x 2 + x 4 2 ! x 6 3 ! + + ( −1 ) n x 2 n n ! + ) d x = C + x x 3 3 + x 5 5.2 ! x 7 7.3 ! + + ( −1 ) n x 2 n + 1 ( 2 n + 1 ) n ! + .
  2. Using the result from part a. we have
    0 1 e x 2 d x = 1 1 3 + 1 10 1 42 + 1 216 .

    The sum of the first four terms is approximately 0.74 . By the alternating series test, this estimate is accurate to within an error of less than 1 216 0.0046296 < 0.01 .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
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 2

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