(iii) (Yk, Yk+1) Approaching either (∞, 0) or (-∞, -∞) Let Yk+1 = = p(yk)", (2.154) where and r are constants, and r > 1, be the desired asymptote to yk+1 = f (yk). Consider the situation where r = 1, p> 1. (2.155) The first approximation is Yk = Ap, A= arbitrary constant. (2.156) Let t = Ap*, z(t) = Yk• (2.157) Therefore, yk+1 = f(yk) becomes z(pt) = f[z(t)], (2.158) and the asymptotic expansion takes the form A1 z(t) =t+ Ao + A2 (2.159) . .. Now let r > 1. Therefore, the first approximation is Yk = (2.160) where A is an arbitrary constant and C = eA > 1. With the definitions t = Cr*, z(t) = Yk; (2.161)
(iii) (Yk, Yk+1) Approaching either (∞, 0) or (-∞, -∞) Let Yk+1 = = p(yk)", (2.154) where and r are constants, and r > 1, be the desired asymptote to yk+1 = f (yk). Consider the situation where r = 1, p> 1. (2.155) The first approximation is Yk = Ap, A= arbitrary constant. (2.156) Let t = Ap*, z(t) = Yk• (2.157) Therefore, yk+1 = f(yk) becomes z(pt) = f[z(t)], (2.158) and the asymptotic expansion takes the form A1 z(t) =t+ Ao + A2 (2.159) . .. Now let r > 1. Therefore, the first approximation is Yk = (2.160) where A is an arbitrary constant and C = eA > 1. With the definitions t = Cr*, z(t) = Yk; (2.161)
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Explain the determaine
![(iii) (Yk, Yk+1) Approaching either (o, 0) or (-∞, -∞)
Let
Yk+1 = p(yk)",
(2.154)
where
and r are constants, and r > 1, be the desired asymptote to yk+1 =
f (yk).
Consider the situation where
r = 1, p> 1.
(2.155)
The first approximation is
Yk = Apk, A = arbitrary constant.
(2.156)
Let
t = Ap", z(t) = Yk•
(2.157)
Therefore, Yk+1
f(yk) becomes
2(pt) = f[z(t)],
(2.158)
and the asymptotic expansion takes the form
A2
A1
t2
z(t) = t+ Ao +
(2.159)
t
Now let r > 1. Therefore, the first approximation is
( Yk = p'/(1-r)eAr* = p!/(1=r)Cn*,
(2.160)
where A is an arbitrary constant and C = eA > 1. With the definitions
t = Crk, z(t) = Yk,
(2.161)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F95cf9130-510f-4366-82e5-bf46cd56ed6f%2Fbebf4e26-129e-43c4-9ed4-2a9a2a2ef989%2Fv1z6iwl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(iii) (Yk, Yk+1) Approaching either (o, 0) or (-∞, -∞)
Let
Yk+1 = p(yk)",
(2.154)
where
and r are constants, and r > 1, be the desired asymptote to yk+1 =
f (yk).
Consider the situation where
r = 1, p> 1.
(2.155)
The first approximation is
Yk = Apk, A = arbitrary constant.
(2.156)
Let
t = Ap", z(t) = Yk•
(2.157)
Therefore, Yk+1
f(yk) becomes
2(pt) = f[z(t)],
(2.158)
and the asymptotic expansion takes the form
A2
A1
t2
z(t) = t+ Ao +
(2.159)
t
Now let r > 1. Therefore, the first approximation is
( Yk = p'/(1-r)eAr* = p!/(1=r)Cn*,
(2.160)
where A is an arbitrary constant and C = eA > 1. With the definitions
t = Crk, z(t) = Yk,
(2.161)
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