7.The Legendre equation of order α is (1−x2)y′′−2xy′+α(α+1)y=0.1−x2y″−2xy′+αα+1y=0. The solution of this equation near the ordinary point x = 0 was discussed in Problems 17 and 18 of Section 5.3. In Example 4 of Section 5.4, it was shown that x = ± 1 are regular singular points. a.Determine the indicial equation and its roots for the point x = 1. b.Find a series solution in powers of x − 1 for x − 1 > 0. Hint: Write 1 + x = 2 + (x − 1) and x = 1 + (x − 1).Alternatively, make the change of variable x − 1 = t and determine a series solution in powers of t.
7.The Legendre equation of order α is (1−x2)y′′−2xy′+α(α+1)y=0.1−x2y″−2xy′+αα+1y=0. The solution of this equation near the ordinary point x = 0 was discussed in Problems 17 and 18 of Section 5.3. In Example 4 of Section 5.4, it was shown that x = ± 1 are regular singular points. a.Determine the indicial equation and its roots for the point x = 1. b.Find a series solution in powers of x − 1 for x − 1 > 0. Hint: Write 1 + x = 2 + (x − 1) and x = 1 + (x − 1).Alternatively, make the change of variable x − 1 = t and determine a series solution in powers of t.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 10CR
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7.The Legendre equation of order α is
(1−x2)y′′−2xy′+α(α+1)y=0.1−x2y″−2xy′+αα+1y=0.
The solution of this equation near the ordinary point x = 0 was discussed in Problems 17 and 18 of Section 5.3. In Example 4 of Section 5.4, it was shown that x = ± 1 are regular singular points.
a.Determine the indicial equation and its roots for the point x = 1.
b.Find a series solution in powers of x − 1 for x − 1 > 0.
Hint: Write 1 + x = 2 + (x − 1) and x = 1 + (x − 1).
Alternatively, make the change of variable x − 1 = t and determine a series solution in powers of t.
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(a) Given
VIEWAdd and subtract 1 in the left side of the Legendre equation as shown below:
VIEWStep 3
VIEWStep 4
VIEWIn order to adjust the index of the second and fourth summations so that x in them has exponent r+n
VIEWCombining the terms as follows:
VIEW(b) The corresponding recurrence relation can be written as follow:
VIEWStep 8
VIEWStep 9
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