1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
jeka94
3 years ago
5

Xy''+2y'-xy by frobenius method

Mathematics
1 answer:
aalyn [17]3 years ago
3 0
First note that x=0 is a regular singular point; in particular x=0 is a pole of order 1 for \dfrac2x.

We seek a solution of the form

y=\displaystyle\sum_{n\ge0}a_nx^{n+r}

where r is to be determined. Differentiating, we have

y'=\displaystyle\sum_{n\ge0}(n+r)a_nx^{n+r-1}
y''=\displaystyle\sum_{n\ge0}(n+r)(n+r-1)a_nx^{n+r-2}

and substituting into the ODE gives

\displaystyle x\sum_{n\ge0}(n+r)(n+r-1)a_nx^{n+r-2}+2\sum_{n\ge0}(n+r)a_nx^{n+r-1}-x\sum_{n\ge0}a_nx^{n+r}=0
\displaystyle \sum_{n\ge0}(n+r)(n+r-1)a_nx^{n+r-1}+2\sum_{n\ge0}(n+r)a_nx^{n+r-1}-\sum_{n\ge0}a_nx^{n+r+1}=0
\displaystyle \sum_{n\ge0}(n+r)(n+r+1)a_nx^{n+r-1}-\sum_{n\ge0}a_nx^{n+r+1}=0
\displaystyle r(r+1)a_0x^{r-1}+(r+1)(r+2)a_1x^r+\sum_{n\ge2}(n+r)(n+r+1)a_nx^{n+r-1}-\sum_{n\ge0}a_nx^{n+r+1}=0
\displaystyle r(r+1)a_0x^{r-1}+(r+1)(r+2)a_1x^r+\sum_{n\ge2}(n+r)(n+r+1)a_nx^{n+r-1}-\sum_{n\ge2}a_{n-2}x^{n+r-1}=0
\displaystyle r(r+1)a_0x^{r-1}+(r+1)(r+2)a_1x^r+\sum_{n\ge2}\bigg((n+r)(n+r+1)a_n-a_{n-2}\bigg)x^{n+r-1}=0

The indicial polynomial, r(r+1), has roots at r=0 and r=-1. Because these roots are separated by an integer, we have to be a bit more careful, but we'll get back to this later.

When r=0, we have the recurrence

a_n=\dfrac{a_{n-2}}{(n+1)(n)}

valid for n\ge2. When n=2k, with k\in\{0,1,2,3,\ldots\}, we find

a_0=a_0
a_2=\dfrac{a_0}{3\cdot2}=\dfrac{a_0}{3!}
a_4=\dfrac{a_2}{5\cdot4}=\dfrac{a_0}{5!}
a_6=\dfrac{a_4}{7\cdot6}=\dfrac{a_0}{7!}

and so on, with a general pattern of

a_{n=2k}=\dfrac{a_0}{(2k+1)!}

Similarly, when n=2k+1 for k\in\{0,1,2,3,\ldots\}, we find

a_1=a_1
a_3=\dfrac{a_1}{4\cdot3}=\dfrac{2a_1}{4!}
a_5=\dfrac{a_3}{6\cdot5}=\dfrac{2a_1}{6!}
a_7=\dfrac{a_5}{8\cdot7}=\dfrac{2a_1}{8!}

and so on, with the general pattern

a_{n=2k+1}=\dfrac{2a_1}{(2k+2)!}

So the first indicial root admits the solution

y=\displaystyle a_0\sum_{k\ge0}\frac{x^{2k}}{(2k+1)!}+a_1\sum_{k\ge0}\frac{x^{2k+1}}{(2k+2)!}
y=\displaystyle \frac{a_0}x\sum_{k\ge0}\frac{x^{2k+1}}{(2k+1)!}+\frac{a_1}x\sum_{k\ge0}\frac{x^{2k+2}}{(2k+2)!}
y=\displaystyle \frac{a_0}x\sum_{k\ge0}\frac{x^{2k+1}}{(2k+1)!}+\frac{a_1}x\sum_{k\ge0}\frac{x^{2k+2}}{(2k+2)!}

which you can recognize as the power series for \dfrac{\sinh x}x and \dfrac{\cosh x}x.

To be more precise, the second series actually converges to \dfrac{\cosh x-1}x, which doesn't satisfy the ODE. However, remember that the indicial equation had two roots that differed by a constant. When r=-1, we may seek a second solution of the form

y=cy_1\ln x+x^{-1}\displaystyle\sum_{n\ge0}b_nx^n

where y_1=\dfrac{\sinh x+\cosh x-1}x. Substituting this into the ODE, you'll find that c=0, and so we're left with

y=x^{-1}\displaystyle\sum_{n\ge0}b_nx^n
y=\dfrac{b_0}x+b_1+b_2x+b_3x^2+\cdots

Expanding y_1, you'll see that all the terms x^n with n\ge0 in the expansion of this new solutions are already accounted for, so this new solution really only adds one fundamental solution of the form y_2=\dfrac1x. Adding this to y_1, we end up with just \dfrac{\sinh x+\cosh x}x.

This means the general solution for the ODE is

y=C_1\dfrac{\sinh x}x+C_2\dfrac{\cosh x}x
You might be interested in
What is the degree of the polynomial <br> F(x)=2x^3-x^2+5x-3
Temka [501]

the degree is the largest exponent on the variable, so it would be 3-

3 0
3 years ago
Four teams will receive prize money based on their ticket sales. The total prize amount to be apportioned is $63. Determine the
Oxana [17]

Answer:

32.43

Step-by-step explanation:

To get the standard divisor

We sum up the tickets sold

Applejack's = 66 tickets

Broncos = 769 tickets

Colts = 652 tickets

Deers = 556 tickets

The total prize amount = $63

Total tickets = 66 + 769 + 652 + 556 = 2043

Standard divisor = 2043/63

= 32.43

The standard divisor to two decimal places is 32.43

8 0
3 years ago
OMG IM SO CONFUSED HELPPPP
Sunny_sXe [5.5K]

Answer:

5.4 is rational. 5.4 divide by 2 is 2.45 5.3 and 5 3 are irrational, as it cant be divided easily

8 0
3 years ago
Two parallel lines are crossed by a transversal. Horizontal and parallel lines b and c are cut by transversal a. At the intersec
pogonyaev

Answer:

x=22

Step-by-step explanation:

According to the Alternate Interior angles theorem, the bottom left angle and the upper right angle have the same measure. Because they were formed from a pair of parallel lines crossed by a transversal one.

So, algebraically:

5x+5=115\\ 5x+5-5=115-5\\ 5x=110\\ \frac{5x}{5}=\frac{115}{5}\\x=22

8 0
3 years ago
Read 2 more answers
The equation y=123x describes the function for the number of toys, y, produced at Toys Plus in x mintues of production time. If
frozen [14]

Answer:

I dont know what the time is but multiply the minutes by 123 and then you get the amount of toys made in that time

4 0
3 years ago
Other questions:
  • What are the solutions of x^2 + 121 = 0?
    13·2 answers
  • A student found the slope of a line that passes through the points (3,15) and (5,9) to be -1/3. what mistake did he make
    15·1 answer
  • If Linnia Bought some flats for flowers and each flat hold 9 flowers, and she has planted 10 flowers, would 9x + 10 be a reasona
    15·1 answer
  • Simplify (4x3+ 13x − 7) − (6x2+ 9x + 2).
    12·1 answer
  • Reid's Hardware discounts all riding lawnmowers 6% to customers paying in cash. If Trey paid $1,278.75 in cash for a riding lawn
    9·1 answer
  • Kurtosis of a normal data distribution is a ___________________ Group of answer choices Measure of data centrality Measure of da
    7·1 answer
  • A number divided by 48 has a quotient of 7 with a remanded of 12.What is the number?
    10·2 answers
  • HELP PLEASE ILL GIVE BRAINLIEST
    15·2 answers
  • Wendell would like to make some chocolate chips cookies. His recipe calls for the use of 1/2 cups of sugar per batch.he has 3 1/
    10·1 answer
  • What is the length of the rectangular plot of land​ shown? Use pencil and paper. How are the lengths of the legs of a right tria
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!