Consider the differential equation (x²-1) y-6x y'+12y-0 where prime denotes the derivative with respect to x. Assume a solution of the form y= c x. л-0 Also assume that the following initial conditions are given y (0) -1, and y'(0) -2. Choose All Correct Answers Below A This equation has no singular points. A particular solution is given by y=x4+2x³+6x²+2x+1 B
Consider the differential equation (x²-1) y-6x y'+12y-0 where prime denotes the derivative with respect to x. Assume a solution of the form y= c x. л-0 Also assume that the following initial conditions are given y (0) -1, and y'(0) -2. Choose All Correct Answers Below A This equation has no singular points. A particular solution is given by y=x4+2x³+6x²+2x+1 B
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
![Consider the differential equation
\[
(x^2 - 1) y'' - 6x y' + 12y = 0
\]
where prime denotes the derivative with respect to \(x\).
Assume a solution of the form \(y = \sum_{n=0}^{\infty} c_n x^n\).
Also, assume that the following initial conditions are given: \(y(0) = 1\), and \(y'(0) = 2\).
**Choose All Correct Answers Below**
A. This equation has no singular points.
B. A particular solution is given by \(y = x^4 + 2x^3 + 6x^2 + 2x + 1\).
C. \(x = 0\) is an ordinary point.
D. A particular solution is given by \(y = x^4 - 2x^3 + 6x^2 - 2x + 1\).
E. \(x = 1\) is an ordinary point.
F. \(x = -1\) is a singular point.
G. The recurrence relation is given by:
\[
c_{n+2} = \frac{(n-2)(n-3)}{(n+2)(n+3)} c_n, \quad n = 0, 1, 2, \ldots
\]
H. The recurrence relation is given by:
\[
c_{n+2} = \frac{(n-3)(n-4)}{(n+1)(n+2)} c_n, \quad n = 0, 1, 2, \ldots
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Transcribed Image Text:Consider the differential equation
\[
(x^2 - 1) y'' - 6x y' + 12y = 0
\]
where prime denotes the derivative with respect to \(x\).
Assume a solution of the form \(y = \sum_{n=0}^{\infty} c_n x^n\).
Also, assume that the following initial conditions are given: \(y(0) = 1\), and \(y'(0) = 2\).
**Choose All Correct Answers Below**
A. This equation has no singular points.
B. A particular solution is given by \(y = x^4 + 2x^3 + 6x^2 + 2x + 1\).
C. \(x = 0\) is an ordinary point.
D. A particular solution is given by \(y = x^4 - 2x^3 + 6x^2 - 2x + 1\).
E. \(x = 1\) is an ordinary point.
F. \(x = -1\) is a singular point.
G. The recurrence relation is given by:
\[
c_{n+2} = \frac{(n-2)(n-3)}{(n+2)(n+3)} c_n, \quad n = 0, 1, 2, \ldots
\]
H. The recurrence relation is given by:
\[
c_{n+2} = \frac{(n-3)(n-4)}{(n+1)(n+2)} c_n, \quad n = 0, 1, 2, \ldots
\]
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