**Problem Statement:**Without actually solving the given differential equation, find the minimum radius of convergence $ R $ of power series solutions about the ordinary point $ x = 0 $.Differential Equation:\[(x^2 - 36)y'' + 7xy' + y = 0\]*About the Ordinary Point $ x = 1 $:*- $ R = (x = 0) $- $ R = (x = 1) $**Explanation:**To find the radius of convergence $ R $ for the power series solution about $ x = 0 $, we identify singular points of the differential equation. For:\[(x^2 - 36)y'' + 7xy' + y = 0\]The singular points occur where the coefficient of the highest derivative, $ (x^2 - 36) $, is zero. Solving:\[x^2 - 36 = 0 \Rightarrow x = \pm 6\]The distance from the ordinary point $ x = 0 $ to the nearest singular point is $ 6 $. Therefore, the minimum radius of convergence $ R $ is $ 6 $.*For the Ordinary Point $ x = 1 $:*Repeating the same procedure, we calculate the distance from $ x = 1 $ to the nearest singular point:- Distance to $ x = 6 $ is $ 6 - 1 = 5 $.- Distance to $ x = -6 $ is $ 6 - (-6) = 7 $.Thus, the minimum radius of convergence $ R $ about $ x = 1 $ is $ 5 $.

The differential equation isThe singular points arethe minimum radius of convergence for…

Answered: Without actually solving the given…

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

**Problem Statement:**

Without actually solving the given differential equation, find the minimum radius of convergence $ R $ of power series solutions about the ordinary point $ x = 0 $.

Differential Equation:
\[
(x^2 - 36)y'' + 7xy' + y = 0
\]

*About the Ordinary Point $ x = 1 $:*
- $ R = (x = 0) $
- $ R = (x = 1) $

**Explanation:**

To find the radius of convergence $ R $ for the power series solution about $ x = 0 $, we identify singular points of the differential equation. For:
\[
(x^2 - 36)y'' + 7xy' + y = 0
\]

The singular points occur where the coefficient of the highest derivative, $ (x^2 - 36) $, is zero. Solving:
\[
x^2 - 36 = 0 \Rightarrow x = \pm 6
\]

The distance from the ordinary point $ x = 0 $ to the nearest singular point is $ 6 $. Therefore, the minimum radius of convergence $ R $ is $ 6 $.

*For the Ordinary Point $ x = 1 $:*

Repeating the same procedure, we calculate the distance from $ x = 1 $ to the nearest singular point:
- Distance to $ x = 6 $ is $ 6 - 1 = 5 $.
- Distance to $ x = -6 $ is $ 6 - (-6) = 7 $.

Thus, the minimum radius of convergence $ R $ about $ x = 1 $ is $ 5 $.

Expert Solution

Step 1: For x=0

The differential equation is

$left parenthesis x squared minus 36 right parenthesis y apostrophe apostrophe plus 7 x y apostrophe plus y equals 0$

The singular points are

$x squared minus 36 equals 0 x equals plus-or-minus 6 L e t space s subscript 1 equals 6 space a n d space s subscript 2 equals negative 6 a n d space o r d i n a r y space p o i n t s space a r e space x subscript 1 equals 0 space a n d space x subscript 2 equals 1$

the minimum radius of convergence for x₁=0

$r subscript 1 equals vertical line s subscript 1 minus x subscript 1 vertical line equals vertical line 6 minus 0 vertical line equals 6 a n d space r subscript 2 equals vertical line s subscript 2 minus x subscript 1 vertical line equals vertical line minus 6 minus 0 vertical line equals 6$

Therefore, the minimum radius of convergence R of power series solutions about the ordinary point x=0 is R= 6.