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The differential equation (x + 1)²y" + (x − 1)(x + 1)y' + 2(x + 2) y = 0 has a series solution of the form ∞ \n+r y = Σan(x − xo)¹+r n=0 about the regular singular point xo = -1, where r is a constant, an are constants and ao 0. (i) Determine the indicial equation, the roots of the indicial equation and the recurrence relation, showing that for n > 1 (n + r − 2)(n+r− 1)an = −(n+r+ 1)an-1. (ii) Show that the series solution corresponding to the larger root of the indicial equation can be written as ∞ y2 = ΑΣ(-1)", n=0 (n + 3)! n! (n + 1)! √(x + 1)n +², where A is an arbitrary constant. Show that this series is convergent and find the radius of convergence for the series.

The differential equation (x + 1)²y" + (x − 1)(x + 1)y' + 2(x + 2) y = 0 has a series solution of the form ∞ \n+r y = Σan(x − xo)¹+r n=0 about the regular singular point xo = -1, where r is a constant, an are constants and ao 0. (i) Determine the indicial equation, the roots of the indicial equation and the recurrence relation, showing that for n > 1 (n + r − 2)(n+r− 1)an = −(n+r+ 1)an-1. (ii) Show that the series solution corresponding to the larger root of the indicial equation can be written as ∞ y2 = ΑΣ(-1)", n=0 (n + 3)! n! (n + 1)! √(x + 1)n +², where A is an arbitrary constant. Show that this series is convergent and find the radius of convergence for the series.

Advanced Engineering Mathematics
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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The differential equation (x + 1)²y" + (x − 1)(x + 1)y' + 2(x + 2) y = 0 has a series solution of the form ∞ \n+r y = Σan(x − xo)¹+r n=0 about the regular singular point xo = -1, where r is a constant, an are constants and ao 0. (i) Determine the indicial equation, the roots of the indicial equation and the recurrence relation, showing that for n ≥ 1 (n + r − 2)(n+r− 1)an = −(n+r+ 1)an-1. (ii) Show that the series solution corresponding to the larger root of the indicial equation can be written as ∞ y2 = ΑΣ(-1)", ; (x + 1)n +², (n + 3)! n! (n + 1)! n=0 where A is an arbitrary constant. Show that this series is convergent and find the radius of convergence for the series.
Transcribed Image Text:The differential equation (x + 1)²y" + (x − 1)(x + 1)y' + 2(x + 2) y = 0 has a series solution of the form ∞ \n+r y = Σan(x − xo)¹+r n=0 about the regular singular point xo = -1, where r is a constant, an are constants and ao 0. (i) Determine the indicial equation, the roots of the indicial equation and the recurrence relation, showing that for n ≥ 1 (n + r − 2)(n+r− 1)an = −(n+r+ 1)an-1. (ii) Show that the series solution corresponding to the larger root of the indicial equation can be written as ∞ y2 = ΑΣ(-1)", ; (x + 1)n +², (n + 3)! n! (n + 1)! n=0 where A is an arbitrary constant. Show that this series is convergent and find the radius of convergence for the series.
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