(a) Show that the power series f(x) = ao (-1)" 22 (n!)2 2²n 71=0 is a general solution to the Bessel equation of order zero. (Hint: It may be easier to write out the first 4-ish terms of f(x) and then take derivative of these terms as opposed to using the summation notation.) (b) Take a power series ansatz for f(x). Then, using that ansatz in the ODE, show how you arrive at the equality Σn(n-1)a+na," + [ ax2=0 71=0 7=2 n=1 (c) By expanding and grouping the summations in (b) by like terms, show that you now arrive at the condition a₁x + [n²a₂ + an 2] 2¹ = 0. n=2 (d) Match the linear, quadratic, cubic, and quartic terms on either side of the equality from (c). You do not need to solve for anything yet, simply match terms. Here is an example of what I mean by matching terms: If I have the equation 2x+3x² = (a + b)x+ (2b+c)x² then by matching my linear terms, I find a + b = 2 and by matching my quadratic terms, I find 2b+c=3. (e) By the matching of terms from (d), argue that all the odd terms of the series for your solution must be 0 (that is, a₁ = Az = A5 = ··· = 0). Do not use the given function in (a) as your reasoning! (f) Still by matching terms, show that a2 = -ao/4. Continue solving for the even terms a4, a6, etc. as a function of an until you see a pattern. By studying this pattern, find a general form for a2n. (Hint: If you do everything correctly, you should start to see the same coefficients as you did in part (a)!)
(a) Show that the power series f(x) = ao (-1)" 22 (n!)2 2²n 71=0 is a general solution to the Bessel equation of order zero. (Hint: It may be easier to write out the first 4-ish terms of f(x) and then take derivative of these terms as opposed to using the summation notation.) (b) Take a power series ansatz for f(x). Then, using that ansatz in the ODE, show how you arrive at the equality Σn(n-1)a+na," + [ ax2=0 71=0 7=2 n=1 (c) By expanding and grouping the summations in (b) by like terms, show that you now arrive at the condition a₁x + [n²a₂ + an 2] 2¹ = 0. n=2 (d) Match the linear, quadratic, cubic, and quartic terms on either side of the equality from (c). You do not need to solve for anything yet, simply match terms. Here is an example of what I mean by matching terms: If I have the equation 2x+3x² = (a + b)x+ (2b+c)x² then by matching my linear terms, I find a + b = 2 and by matching my quadratic terms, I find 2b+c=3. (e) By the matching of terms from (d), argue that all the odd terms of the series for your solution must be 0 (that is, a₁ = Az = A5 = ··· = 0). Do not use the given function in (a) as your reasoning! (f) Still by matching terms, show that a2 = -ao/4. Continue solving for the even terms a4, a6, etc. as a function of an until you see a pattern. By studying this pattern, find a general form for a2n. (Hint: If you do everything correctly, you should start to see the same coefficients as you did in part (a)!)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
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![(a) Show that the power series
\[
f(x) = a_0 \sum_{n=0}^{\infty} \frac{(-1)^n}{2^{2n}(n!)^2}x^{2n},
\]
is a general solution to the Bessel equation of order zero. *(Hint: It may be easier to write out the first 4-ish terms of \( f(x) \) and then take derivative of these terms as opposed to using the summation notation.)*
(b) Take a power series ansatz for \( f(x) \). Then, using that ansatz in the ODE, show how you arrive at the equality
\[
\sum_{n=2}^{\infty} n(n-1)a_nx^{n-2} + \sum_{n=1}^{\infty} na_nx^{n-2} + \sum_{n=0}^{\infty} a_nx^{n+2} = 0
\]
(c) By expanding and grouping the summations in (b) by like terms, show that you now arrive at the condition
\[
a_1 x + \sum_{n=2}^{\infty} [n^2a_n + a_{n-2}] x^{n} = 0.
\]
(d) Match the linear, quadratic, cubic, and quartic terms on either side of the equality from (c). You do not need to solve for anything yet, simply match terms. Here is an example of what I mean by matching terms:
If I have the equation
\[
2x + 3x^2 = (a + b)x + (2b + c)x^2
\]
then by matching my linear terms, I find \( a + b = 2 \) and by matching my quadratic terms, I find \( 2b + c = 3 \).
(e) By the matching of terms from (d), argue that all the odd terms of the series for your solution must be 0 (that is, \( a_1 = a_3 = a_5 = \ldots = 0 \)). Do not use the given function in (a) as your reasoning!
(f) Still by matching terms, show that \( a_2 = -a_0/4 \). Continue solving for the even terms \( a_4, a](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7bf14c43-8cf8-40bd-94f2-90056f6925f0%2Fb1a20d49-7db1-4a3b-8025-5f3356893cdd%2Fyvbwkql_processed.png&w=3840&q=75)
Transcribed Image Text:(a) Show that the power series
\[
f(x) = a_0 \sum_{n=0}^{\infty} \frac{(-1)^n}{2^{2n}(n!)^2}x^{2n},
\]
is a general solution to the Bessel equation of order zero. *(Hint: It may be easier to write out the first 4-ish terms of \( f(x) \) and then take derivative of these terms as opposed to using the summation notation.)*
(b) Take a power series ansatz for \( f(x) \). Then, using that ansatz in the ODE, show how you arrive at the equality
\[
\sum_{n=2}^{\infty} n(n-1)a_nx^{n-2} + \sum_{n=1}^{\infty} na_nx^{n-2} + \sum_{n=0}^{\infty} a_nx^{n+2} = 0
\]
(c) By expanding and grouping the summations in (b) by like terms, show that you now arrive at the condition
\[
a_1 x + \sum_{n=2}^{\infty} [n^2a_n + a_{n-2}] x^{n} = 0.
\]
(d) Match the linear, quadratic, cubic, and quartic terms on either side of the equality from (c). You do not need to solve for anything yet, simply match terms. Here is an example of what I mean by matching terms:
If I have the equation
\[
2x + 3x^2 = (a + b)x + (2b + c)x^2
\]
then by matching my linear terms, I find \( a + b = 2 \) and by matching my quadratic terms, I find \( 2b + c = 3 \).
(e) By the matching of terms from (d), argue that all the odd terms of the series for your solution must be 0 (that is, \( a_1 = a_3 = a_5 = \ldots = 0 \)). Do not use the given function in (a) as your reasoning!
(f) Still by matching terms, show that \( a_2 = -a_0/4 \). Continue solving for the even terms \( a_4, a
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