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
Aleonysh [2.5K]
3 years ago
9

Find two linearly independent solutions to the equation y"-2xy'+2y=0 in the form of a power series.

Mathematics
1 answer:
ioda3 years ago
6 0

We want a solution in the form

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

with derivatives

y'=\displaystyle\sum_{n\ge0}(n+1)a_{n+1}x^n

y''=\displaystyle\sum_{n\ge0}(n+2)(n+1)a_{n+2}x^n

Substituting y and its derivatives into the ODE,

y''-2xy'+2y=0

gives

\displaystyle\sum_{n\ge0}(n+2)(n+1)a_{n+2}x^n-2\sum_{n\ge0}(n+1)a_{n+1}x^{n+1}+2\sum_{n\ge0}a_nx^n=0

Shift the index on the second sum to have it start at n=1:

\displaystyle\sum_{n\ge0}(n+1)a_{n+1}x^{n+1}=\sum_{n\ge1}na_nx^n

and take the first term out of the other two sums. Then we can consolidate the sums into one that starts at n=1:

\displaystyle(2a_2+2a_0)+\sum_{n\ge1}\bigg[(n+2)(n+1)a_{n+2}+(2-2n)a_n\bigg]x^n=0

and so the coefficients in the series solution are given by the recurrence,

\begin{cases}a_0=y(0)\\a_1=y'(0)\\(n+2)(n+1)a_{n+2}=2(n-1)a_n&\text{for }n\ge0\end{cases}

or more simply, for n\ge2,

a_n=\dfrac{2(n-3)}{n(n-1)}a_{n-2}

Note the dependency between every other coefficient. Consider the two cases,

  • If n=2k, where k\ge0 is an integer, then

k=0\implies n=0\implies a_0=a_0

k=1\implies n=2\implies a_2=-a_0=2^1\dfrac{(-1)}{2!}a_0

k=2\implies n=4\implies a_4=\dfrac{2\cdot1}{4\cdot3}a_2=2^2\dfrac{1\cdot(-1)}{4!}a_0

k=3\implies n=6\implies a_6=\dfrac{2\cdot3}{6\cdot5}a_4=2^3\dfrac{3\cdot1\cdot(-1)}{6!}a_0

k=4\implies n=8\implies a_8=\dfrac{2\cdot5}{8\cdot7}a_6=2^4\dfrac{5\cdot3\cdot1\cdot(-1)}{8!}a_0

and so on, with the general pattern

a_{2k}=\dfrac{2^ka_0}{(2k)!}\displaystyle\prod_{i=1}^k(2i-3)

  • If n=2k+1, then

k=0\implies n=1\implies a_1=a_1

k=1\implies n=3\implies a_3=\dfrac{2\cdot0}{3\cdot2}a_1=0

and we would see that a_{2k+1}=0 for all k\ge1.

So we have

y(x)=\displaystyle\sum_{k\ge0}\bigg[a_{2k}x^{2k}+a_{2k+1}x^{2k+1}\bigg]

so that one solution is

\boxed{y_1(x)=\displaystyle a_0\sum_{k\ge0}\frac{2^k\prod\limits_{i=1}^k(2i-3)}{(2k)!}x^{2k}}

and the other is

\boxed{y_2(x)=a_1x}

I've attached a plot of the exact and series solutions below with a_0=y(0)=1, a_1=y'(0)=1, and 0\le k\le5 to demonstrate that the series solution converges to the exact one.

You might be interested in
find the amount and the compound interest on rupees 12000 at 6% p.a. compounded half yearly for 3/2 years ​
stiv31 [10]

Answer:

A= ₹3,112.72

CI = ₹1,112.72

Step-by-step explanation:

Here P = ₹ 12000

Since, interest is compounded half yearly.

Therefore, R = 6/2 = 3%

n = 3/2* 2 = 3 half years

\because A= P\bigg(1+\frac{R}{100}\bigg)^n

\therefore A= 12000\times \bigg(1+\frac{3}{100}\bigg)^3

\therefore A= 12000\times \bigg(\frac{100+3}{100}\bigg)^3

\therefore A= 12000\times \bigg(\frac{103}{100}\bigg)^3

\therefore A= 12000\times \bigg(1.03\bigg)^3

\therefore A= 12000\times 1.092727

\therefore A= 13,112.724

\therefore A=Rs. \:13,112.72

CI = A - P

CI = 13,112.72 - 12000

CI = ₹1,112.72

6 0
3 years ago
What is 0.01234567901 as a fraction?
nirvana33 [79]
12,345,678,901/100,000,000,000
3 0
3 years ago
If a student wanted to immediately eliminate (get rid of) the fraction in this problem, they should:
mash [69]

Answer:

2x-15=12

Step-by-step explanation:

4 0
3 years ago
Is this graph a function or not a function?
weeeeeb [17]

Answer:

It is a function.

Step-by-step explanation:

You can test if a graph is a function if you draw a vertical line anywhere on the graph and you see it hits two points.

This is the table for the graph.

\left[\begin{array}{ccc}x&y\\-3&0\\0&1\\3&2\end{array}\right]

Remember these rules:

  • Each x value, or input, has its unique y value, or output
  • If you draw a vertical line anywhere on the graph, it should only go through one point

We can check these two rules for this graph:

  • Does each x value have its own, unique y value? Yes
  • If you draw a vertical line anywhere on the graph, does it only go through one point? Yes, there are no overlaps

Keep in mind that two different x-values can have the same y value.

Figure 1:

It has two x values with the same y-values.

Figure 2 and 3:

The vertical line goes through two points. So the same x-value has two different y-values.

-Chetan K

4 0
2 years ago
Sebastian rides his bike 2,000 meters in 5 minutes. How many meters does he bike in 1 minute?
USPshnik [31]

Answer:400 meters

Step-by-step explanation:

2000/5=400

5 0
3 years ago
Read 2 more answers
Other questions:
  • Pete has 5 marbles . Jay has a number of marbles that is two more than 5 . How many marbles does jay have?
    8·2 answers
  • Solve the equation for n.<br> 5 = -4 + 27/n
    6·2 answers
  • There are 10 women and 3 men in room A. One person is picked at random from room A and moved to room B, where there are already
    7·1 answer
  • Y to the power of 3=64
    12·2 answers
  • Which expression is equivalent to 4(u+11w)<br> ? PLZ I need this
    8·1 answer
  • If you ask a question will the points given ne deducted from your points​
    15·2 answers
  • What expression is equivalent to x^2-7x+12+(x-3)^2
    9·1 answer
  • -28 + 2x = -18<br> x=? Solve for x
    10·2 answers
  • Given the function
    9·1 answer
  • 11. FOUR 4's. Using only four 4's, the operations (+,-, x, +), and parentheses,
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!