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

y = 5 e 2 x

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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 + ) .
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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 )

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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 .
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Questions & Answers

What is scarcity.
Npoanlarb Reply
why our wants are limited
Npoanlarb Reply
nooo want is unlimited but resources are limited
Ruchi
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Madara
our wants are not limited but rather the resources
Moses
as we know that there are two principle of microeconomics scarcity of resources and they have alternative uses...
Ruchi
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what is demand
Thank Reply
demand is something wt we called in economic theory of demand it simply means if price of product is increase then demand of product will decrease
Ruchi
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in microeconomic
Ruchi
demand is what and how much you want and what's your need...
Shikhar
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Chiamaka Reply
because demand is increase
Ruchi
because demand is increase
Patience
but how demand increases?
Aziz
Because of the Marketing and purchasing power of people.
AmarbirSingh
but how could we know that people's demands have increased everyday by day and how could we know that this is time to produced the products in the market. Is any connection among them
yaqoob
for normal good people demand remain the same if price of product will increase or not
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see that some product which increases day by day is comes under normal good which is used by consumer
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Seems hot discussing going here
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If there are less products demand starts to increase for those products
Shamamet
Economics is really interesting to learn ....
Shamamet
see there is Inferior goods ands normal goods inferior good demand is rarely increase whereas as we talk about normal good demand will absolutely Increase whether price is increase or not
Ruchi
and demand for normal goods increase cause people's income as a while increases time to time
Abhisek
and it might also be that the cost of raw materials are high.
ATTAH
may be
Ruchi
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Shikhar
hmmm there is inverse relationship between demand and price
Ruchi
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Odunayomi Reply
the nature and significance of economics studies
Deborah
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Shuaib Reply
deman is amount of goods and services a consumer is willing and able to buy or purchase at a given price.
Sainabou
the willingness and ability of a body to purchase goods nd servicesbis called demand ,so if she/has ability but doesn't have willingness it's not a demand same if she or he has willingness but doesn't has ability it's not a demand too
Gul
Demand refers to as quantities of a goods and services in which consumers are willing and able to purchase at a given period of time and demand can also be defined as the desire or willingness and backed by the ability to pay.
Fadiga
Yeah
Mathias
What is Choice
Kofi
Choice refers to the ability of a consumer or producer to decide which good, service or resource to purchase or provide from a range of possible options. Being free to chose is regarded as a fundamental indicator of economic well being and development.
Shonal
choice is a act of selecting or choosing from the numerous or plenty wants.
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Compare and contract the function of commercial bank and the central bank of Nigeria
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economics is a social science that study's how resources can be used to produce goods and services for society
Nathan
Economic is a science which studies human behavior as a relationship between ends and scares means which have alternatives uses or purposes.
Fadiga
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rules
Buayadarat_Gaming
unlimited wants vs limited resources
Nathan
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Paradigm shift it is the reconcilliation of fedural goods in production
Shyline
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Aziz
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crux
Shyline
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

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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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