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
Ratling [72]
3 years ago
11

y′′ −y = 0, x0 = 0 Seek power series solutions of the given differential equation about the given point x 0; find the recurrence

relation that the coefficients must satisfy
Mathematics
1 answer:
sukhopar [10]3 years ago
4 0

Let

\displaystyle y(x) = \sum_{n=0}^\infty a_nx^n = a_0 + a_1x + a_2x^2 + \cdots

Differentiating twice gives

\displaystyle y'(x) = \sum_{n=1}^\infty na_nx^{n-1} = \sum_{n=0}^\infty (n+1) a_{n+1} x^n = a_1 + 2a_2x + 3a_3x^2 + \cdots

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

When x = 0, we observe that y(0) = a₀ and y'(0) = a₁ can act as initial conditions.

Substitute these into the given differential equation:

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

\displaystyle \sum_{n=0}^\infty \bigg((n+2)(n+1) a_{n+2} - a_n\bigg) x^n = 0

Then the coefficients in the power series solution are governed by the recurrence relation,

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

Since the n-th coefficient depends on the (n - 2)-th coefficient, we split n into two cases.

• If n is even, then n = 2k for some integer k ≥ 0. Then

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

k=1 \implies n=2 \implies a_2 = \dfrac{a_0}{2\cdot1}

k=2 \implies n=4 \implies a_4 = \dfrac{a_2}{4\cdot3} = \dfrac{a_0}{4\cdot3\cdot2\cdot1}

k=3 \implies n=6 \implies a_6 = \dfrac{a_4}{6\cdot5} = \dfrac{a_0}{6\cdot5\cdot4\cdot3\cdot2\cdot1}

It should be easy enough to see that

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

• If n is odd, then n = 2k + 1 for some k ≥ 0. Then

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

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

k = 2 \implies n=5 \implies a_5 = \dfrac{a_3}{5\cdot4} = \dfrac{a_1}{5\cdot4\cdot3\cdot2}

k=3 \implies n=7 \implies a_7=\dfrac{a_5}{7\cdot6} = \dfrac{a_1}{7\cdot6\cdot5\cdot4\cdot3\cdot2}

so that

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

So, the overall series solution is

\displaystyle y(x) = \sum_{n=0}^\infty a_nx^n = \sum_{k=0}^\infty \left(a_{2k}x^{2k} + a_{2k+1}x^{2k+1}\right)

\boxed{\displaystyle y(x) = a_0 \sum_{k=0}^\infty \frac{x^{2k}}{(2k)!} + a_1 \sum_{k=0}^\infty \frac{x^{2k+1}}{(2k+1)!}}

You might be interested in
2y+2y+4y^3+4-3y^3+4+y​
RUDIKE [14]

Answer:

y^3 + 5y + 8

Step-by-step explanation:

Collect like terms

4y^3 - 3y^3 + 2y + 2y + y + 4 + 4

= 1y^3 + 5y + 8 OR y^3 + 5y + 8

4 0
3 years ago
Read 2 more answers
I need help with this question?
tekilochka [14]

Answer:

8.5 h = 510 min

Step-by-step explanation:

An hour is equal to 60 minutes. Half an hour is 30 minutes.

We can break "8.5" into 8 hours and 0.5 of an hour (which is 30 minutes).

Knowing this, we can multiply 8 by 60 since each hour is 60 minutes. We then get 480. We still need to add the remaining 0.5 of an hour (30 minutes) so we add 480 + 30. We then get 510.

Therefore 8.5 hours = 510 minutes. Hope this helps!

5 0
3 years ago
Read 2 more answers
Calcular el volumen de un cono con altura de 15cm y una base cuyo radio es de 3cm
Ganezh [65]

Answer:

v= 141.37

Step-by-step explanation:

π · 3^2 · 15/ 3

=141.37167

6 0
3 years ago
The summer sports training camp has a maximum capacity of 250 students. If 85 have preregistered, then how many students can sti
tangare [24]

Solving by an inequality is : x ≤ 130
6 0
3 years ago
A rectangular park is 5/6 miles wide and 1 5/7 miles long. What is the area of the park?
zysi [14]
1 3/7 or 10/7 I took the test aswell
3 0
3 years ago
Other questions:
  • Two mechanics can assemble 10 bicycles in 8 hours. How many bicycles could one mechanic assemble in 8 hours?
    8·1 answer
  • Solve for x... I dont know what to do with the square root
    6·1 answer
  • A category within biology that deals with the physical and chemical functions.
    6·2 answers
  • The height of a triangle is 2 less than 5 times its base. If the base of the triangle is x feet and the area of the triangle is
    7·1 answer
  • Find the area of the triangles. Show all work and make sure to include the formula you used!
    10·1 answer
  • Tomic and Brooke make a game of shooting free throws. Each basket is worth 2 points. Each miss is worth
    15·2 answers
  • Please help i need it so bad
    12·1 answer
  • HELP ASAP ILL MARK BRAINLIEST
    10·2 answers
  • Which time unit measures a very short time interval?
    10·1 answer
  • A summer camp cookout is planned for the campers and their families. There is room for 200 people. Each adult costs $4, and each
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!