1. Develop series solutions for the Hermite equation a) For k = 0 aj+2 2aj For k = 1 Y Yeven aj+2 odd = = ao 2aj j - a (j+1)(j+2)' 2(-a)x² 2! 1+ j+1-a (j+2) (j + 3)' 2(1 - a)x³ 3! ao x + + + (jeven), 2² (-a)(2-a)x4 4! + (j odd), 2² (1 - a)(3- a)x5 5! 1 + 1 b) Show that both series are convergent for all x, the ratio of successive coefficients behaves for large indices as the corresponding ratio in the expansion of exp(2x²). c) Show that by an appropriate choice of a the series solutions can be cut and converted into finite polynomials.
1. Develop series solutions for the Hermite equation a) For k = 0 aj+2 2aj For k = 1 Y Yeven aj+2 odd = = ao 2aj j - a (j+1)(j+2)' 2(-a)x² 2! 1+ j+1-a (j+2) (j + 3)' 2(1 - a)x³ 3! ao x + + + (jeven), 2² (-a)(2-a)x4 4! + (j odd), 2² (1 - a)(3- a)x5 5! 1 + 1 b) Show that both series are convergent for all x, the ratio of successive coefficients behaves for large indices as the corresponding ratio in the expansion of exp(2x²). c) Show that by an appropriate choice of a the series solutions can be cut and converted into finite polynomials.
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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![1. Develop series solutions for the Hermite equation
a) For k = 0
aj+2
2a,-
For k = 1
Y
Yeven
aj+2 =
odd
=
=
j - a
(j+1)(j+2)
ao 1 +
2aj
2(-a)x²
2!
j+1 - a
(j+2) (j + 3)'
2(1-a)x³
3!
ao x +
+
+
(jeven),
2²(-a)(2 − a)x¹
+
(j odd),
2²(1 — a)(3 — a)x5
5!
+...].
b) Show that both series are convergent for all x, the ratio of successive coefficients behaves for large indices as the
corresponding ratio in the expansion of
exp(2x²).
c) Show that by an appropriate choice of a the series solutions can be cut and converted into finite polynomials.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7536a213-9eec-4280-9405-2460faef03dd%2F75819f41-0efe-49b8-b7f5-13c1a56d0fe5%2Fmi11si8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Develop series solutions for the Hermite equation
a) For k = 0
aj+2
2a,-
For k = 1
Y
Yeven
aj+2 =
odd
=
=
j - a
(j+1)(j+2)
ao 1 +
2aj
2(-a)x²
2!
j+1 - a
(j+2) (j + 3)'
2(1-a)x³
3!
ao x +
+
+
(jeven),
2²(-a)(2 − a)x¹
+
(j odd),
2²(1 — a)(3 — a)x5
5!
+...].
b) Show that both series are convergent for all x, the ratio of successive coefficients behaves for large indices as the
corresponding ratio in the expansion of
exp(2x²).
c) Show that by an appropriate choice of a the series solutions can be cut and converted into finite polynomials.
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