7- y" + xy = 0, 20 = 0 > Answer Solution Let y = ao + ajx + azx²+.+an¤"+... Then y" =n(n – 1)a„x"-² = >(n +2)(n +1)an+2x". n=0 Substitution into the differential equation results in (n+2)(n+ 1)an+2a" + x anx" = 0 n=0 n=0 or 2. la2 +>[(n+2)(n+1)an+2 + an-1]x" = 0. [(n + 2)(n+ 1)an+2+ an-1]a" = 0. n=1 First, a2 = 0. Next, equating all the coefficients to zero, (n + 2)(n +1)a,+2+ an-1 = 0, n= 1,2, ...
7- y" + xy = 0, 20 = 0 > Answer Solution Let y = ao + ajx + azx²+.+an¤"+... Then y" =n(n – 1)a„x"-² = >(n +2)(n +1)an+2x". n=0 Substitution into the differential equation results in (n+2)(n+ 1)an+2a" + x anx" = 0 n=0 n=0 or 2. la2 +>[(n+2)(n+1)an+2 + an-1]x" = 0. [(n + 2)(n+ 1)an+2+ an-1]a" = 0. n=1 First, a2 = 0. Next, equating all the coefficients to zero, (n + 2)(n +1)a,+2+ an-1 = 0, n= 1,2, ...
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
solve the given differential equation by means of a power series about the given point x0. Find the recurrence relation; also find the first four terms in each of two linearly independent solutions (unless the series terminates sooner). If possible, find the general term in each solution.
For this question, I do not understand why we need to separate the 2*a2? And what I got is when n=0, 1, 3, 4, 5=0. For I plug the 0, 1, 3, 4, 5 into the equation an+2=-an-1/(n+2)(n-1)
![7. y" + xy = 0,
xo = 0
Answer
Solution
Let y = ao + a1x + a2x²+...+anx"+.... Then
y" =n(n – 1)a,a"-2 = (n + 2)(n + 1)an+2¤".
n=2
n=0
Substitution into the differential equation results in
>(n + 2)(n + 1)an+2¤" + x >
anx" = 0
n=0
n=0
or
2· la2 +[(n + 2)(n + 1)an+2 + an-1]x" = 0.
n=1
First, a2 =
0. Next, equating all the coefficients to zero,
(n + 2)(n + 1)an+2 + an-1 = 0, n = 1, 2, ...
L072](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F81d7096f-8065-4480-a939-1687e3ea3984%2F07531192-67b1-4e4d-a2ac-50ed3994dbbd%2Fs930gt_processed.png&w=3840&q=75)
Transcribed Image Text:7. y" + xy = 0,
xo = 0
Answer
Solution
Let y = ao + a1x + a2x²+...+anx"+.... Then
y" =n(n – 1)a,a"-2 = (n + 2)(n + 1)an+2¤".
n=2
n=0
Substitution into the differential equation results in
>(n + 2)(n + 1)an+2¤" + x >
anx" = 0
n=0
n=0
or
2· la2 +[(n + 2)(n + 1)an+2 + an-1]x" = 0.
n=1
First, a2 =
0. Next, equating all the coefficients to zero,
(n + 2)(n + 1)an+2 + an-1 = 0, n = 1, 2, ...
L072
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