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
mariarad [96]
3 years ago
5

Power Series Differential equation

Mathematics
1 answer:
KatRina [158]3 years ago
4 0
The next step is to solve the recurrence, but let's back up a bit. You should have found that the ODE in terms of the power series expansion for y

\displaystyle\sum_{n\ge2}\bigg((n-3)(n-2)a_n+(n+3)(n+2)a_{n+3}\bigg)x^{n+1}+2a_2+(6a_0-6a_3)x+(6a_1-12a_4)x^2=0

which indeed gives the recurrence you found,

a_{n+3}=-\dfrac{n-3}{n+3}a_n

but in order to get anywhere with this, you need at least three initial conditions. The constant term tells you that a_2=0, and substituting this into the recurrence, you find that a_2=a_5=a_8=\cdots=a_{3k-1}=0 for all k\ge1.

Next, the linear term tells you that 6a_0+6a_3=0, or a_3=a_0.

Now, if a_0 is the first term in the sequence, then by the recurrence you have

a_3=a_0
a_6=-\dfrac{3-3}{3+3}a_3=0
a_9=-\dfrac{6-3}{6+3}a_6=0

and so on, such that a_{3k}=0 for all k\ge2.

Finally, the quadratic term gives 6a_1-12a_4=0, or a_4=\dfrac12a_1. Then by the recurrence,

a_4=\dfrac12a_1
a_7=-\dfrac{4-3}{4+3}a_4=\dfrac{(-1)^1}2\dfrac17a_1
a_{10}=-\dfrac{7-3}{7+3}a_7=\dfrac{(-1)^2}2\dfrac4{10\times7}a_1
a_{13}=-\dfrac{10-3}{10+3}a_{10}=\dfrac{(-1)^3}2\dfrac{7\times4}{13\times10\times7}a_1

and so on, such that

a_{3k-2}=\dfrac{a_1}2\displaystyle\prod_{i=1}^{k-2}(-1)^{2i-1}\frac{3i-2}{3i+4}

for all k\ge2.

Now, the solution was proposed to be

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

so the general solution would be

y=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5+a_6x^6+\cdots
y=a_0(1+x^3)+a_1\left(x+\dfrac12x^4-\dfrac1{14}x^7+\cdots\right)
y=a_0(1+x^3)+a_1\displaystyle\left(x+\sum_{n=2}^\infty\left(\prod_{i=1}^{n-2}(-1)^{2i-1}\frac{3i-2}{3i+4}\right)x^{3n-2}\right)
You might be interested in
What is the slope of the line that passes through the points (10,5) and (9,5)? Write
Alex_Xolod [135]

Answer:

slope of line is 0.

Step-by-step explanation:

Slope of a line passing through (x1,y1) and (x2,y2) is given by

m = (y2-y1)/(x2-x1)

Given points are  (10,5) and (9,5)

thus

slope = (5-5)/(9-10) = 0/-1 = 0

Thus, slope of line is 0.

When slope of line is zero it means that the line the which passes through given point is parallel to x-axis.

4 0
3 years ago
What is the value of 3-(-2)?
Akimi4 [234]

Answer:

5

Step-by-step explanation:

The double negatives turn into a positive which makes the problem 3+2 which equals 5.

3 0
3 years ago
Alexa buys 27 stamps, each stamp is of 1 square unit of area. She exchanges them for 8 sugar cubes, whose total volume is numeri
Vikentia [17]
The answer is 8 sugar cubes
8 0
3 years ago
Which equation models the cost of the tickets?
Gelneren [198K]
A.


10.50 divided by 2 is 5.25. meaning that each ticket is 5.25
4 0
2 years ago
Find the real or imaginary solutions of 125x^3 - 27 = 0​
hichkok12 [17]

Answer:

1.2}3,2

Step-by-step explanation:

4 0
3 years ago
Other questions:
  • Lisa walked 48 blocks in 3 hours how much blocks per hours
    5·1 answer
  • What comes next in the sequence:<br> 5,13, 29,61,?
    9·2 answers
  • WHAT IS THE RELATIONSHIP BETWEEN THE ANGLE OF ELEVATION AND THE ANGLE OF DEPRESSION? GIVE AN EXAMPLE OF AN APPLICATION USING THE
    12·2 answers
  • The sum of 3 fifteens and 17 fifteens
    12·2 answers
  • Help please please lol
    12·1 answer
  • How to divided 45÷7 need help please
    13·1 answer
  • A truck is said to get 18 miles per gallon on a highway, but this value can fluctuate,
    9·1 answer
  • Use the graph to find the possible value for X such that f(x) =10
    14·2 answers
  • Does the greater than or less than
    5·1 answer
  • What is the inverse of A(r)=15+3r
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!