Please answer this NEATLY, COMPLETELY, and CORRECTLY for an UPVOTE.I already have the final answers (general solutions) for each given. I just need the complete and step-by-step process/solution on how it arrived to the final answer.This topic falls under Power Series. Find the G.S. about x, = 0 for the following DE's(x2 + 4)у" + 2ху' — 12у %3D 0а.(-1)*(k + 1)3a,x2kΣAnswer: y = ao + a,x +75 a1x³ +22k (2k – 3)(2k – 1)k=112(1+ x²)y" + 10xy' + 20y = 0b.(-1)*(2k + 1)(2k + 3) (k + 1)aox2к3(-1)*(k + 1)(k + 2)(2k + 3)a,x2k+1Answer: y = ɑo + a1x + >.k=1k=1c.(x² + 4)y" + xy = x + 21a1 +-x44811- ao2 - aoAnswer: y = >Σ.anx" = ao + a‚x+7x²x5 + •320...24n=0

a) Given thatx2+4y"+2xy'-12y=0 ——(1)Let y=∑n=0∞Cnxny'=∑n=0∞nCnxn-1=y"=∑n=2∞n(n-1)Cnxn-2

Find the G.S. about x, = 0 for the following DE's (x2 + 4)у" + 2ху' — 12у %3D 0 а. (-1)*(k + 1)3a,x2k Σ Answer: y = ao + a,x +75 a1x³ + 22k (2k – 3)(2k – 1) k=1 12 (1+ x²)y" + 10xy' + 20y = 0 b. (-1)*(2k + 1)(2k + 3) (k + 1)aox2к 3 (-1)*(k + 1)(k + 2)(2k + 3)a,x2k+1 Answer: y = ɑo + a1x + >. k=1 k=1 c. (x² + 4)y" + xy = x + 2 1 a1 + -x4 48 1 1- ao 2 - ao Answer: y = > anx" = ao + a‚x +7x² x5 + • 320 ... 24 n=0

Find the G.S. about x, = 0 for the following DE's (x2 + 4)у" + 2ху' — 12у %3D 0 а. (-1)(k + 1)3a,x2k Σ Answer: y = ao + a,x +75 a1x³ + 22k (2k – 3)(2k – 1) k=1 12 (1+ x²)y" + 10xy' + 20y = 0 b. (-1)(2k + 1)(2k + 3) (k + 1)aox2к 3 (-1)*(k + 1)(k + 2)(2k + 3)a,x2k+1 Answer: y = ɑo + a1x + >. k=1 k=1 c. (x² + 4)y" + xy = x + 2 1 a1 + -x4 48 1 1- ao 2 - ao Answer: y = > anx" = ao + a‚x +7x² x5 + • 320 ... 24 n=0

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

Please answer this NEATLY, COMPLETELY, and CORRECTLY for an UPVOTE.

I already have the final answers (general solutions) for each given. I just need the complete and step-by-step process/solution on how it arrived to the final answer.

This topic falls under Power Series.

Find the G.S. about x, = 0 for the following DE's
(x2 + 4)у" + 2ху' — 12у %3D 0
а.
(-1)*(k + 1)3a,x2k
Σ
Answer: y = ao + a,x +75 a1x³ +
22k (2k – 3)(2k – 1)
k=1
12
(1+ x²)y" + 10xy' + 20y = 0
b.
(-1)*(2k + 1)(2k + 3) (k + 1)aox2к
3
(-1)*(k + 1)(k + 2)(2k + 3)a,x2k+1
Answer: y = ɑo + a1x + >.
k=1
k=1
c.
(x² + 4)y" + xy = x + 2
1
a1 +
-x4
48
1
1- ao
2 - ao
Answer: y = >
Σ.
anx" = ao + a‚x+7x²
x5 + •
320
...
24
n=0

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