The second solution may be obtained using Abel's theorem which states that for a second- order ODE of the form y" + P(x)y' + Q(x)y=0 with first solution y₁(x), the second solution is given by I 1 3₂2(x) = y₁1(x) { /** 1² (x²) exp [- [* P{z^")&²"]az"}. P(x")d For the case k = 0, with first solution y₁(x) Jo(a), by keeping leading-order terms only, show that Abel's theorem leads to the second-solution given by Y₂(x) = N₁(x) = Jo(x) [In(x) + O(x²)], where O(1²) collects all terms of second and higher powers of a. Explain in brief why this second solution could not be obtained using the method of Frobe- niu
The second solution may be obtained using Abel's theorem which states that for a second- order ODE of the form y" + P(x)y' + Q(x)y=0 with first solution y₁(x), the second solution is given by I 1 3₂2(x) = y₁1(x) { /** 1² (x²) exp [- [* P{z^")&²"]az"}. P(x")d For the case k = 0, with first solution y₁(x) Jo(a), by keeping leading-order terms only, show that Abel's theorem leads to the second-solution given by Y₂(x) = N₁(x) = Jo(x) [In(x) + O(x²)], where O(1²) collects all terms of second and higher powers of a. Explain in brief why this second solution could not be obtained using the method of Frobe- niu
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
![The second solution may be obtained using Abel's theorem which states that for a second-
order ODE of the form y" + P(x)y' + Q(x)y=0 with first solution y₁(x), the second solution
is given by
1
92(0) = {4} {* }(250p [- SP (25)dhe" ]de"}.
y₁(x)
exp
(x²)
For the case k = 0, with first solution y₁(x) Jo(a), by keeping leading-order terms only,
show that Abel's theorem leads to the second-solution given by
Y₂(x) = N₁(x) = Jo(x) [In(x) + O(x²)],
where O(1²) collects all terms of second and higher powers of a.
Explain in brief why this second solution could not be obtained using the method of Frobe-
niu](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffade91a1-6997-42f9-8725-9969e19df335%2Fcc925c31-3107-4c52-bbc8-9cafceae51b2%2Fup8ky36_processed.png&w=3840&q=75)
Transcribed Image Text:The second solution may be obtained using Abel's theorem which states that for a second-
order ODE of the form y" + P(x)y' + Q(x)y=0 with first solution y₁(x), the second solution
is given by
1
92(0) = {4} {* }(250p [- SP (25)dhe" ]de"}.
y₁(x)
exp
(x²)
For the case k = 0, with first solution y₁(x) Jo(a), by keeping leading-order terms only,
show that Abel's theorem leads to the second-solution given by
Y₂(x) = N₁(x) = Jo(x) [In(x) + O(x²)],
where O(1²) collects all terms of second and higher powers of a.
Explain in brief why this second solution could not be obtained using the method of Frobe-
niu
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