- #1
sonia akram
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what is the indicial equation associated with regular singularities x=1 and x=-1 of legendre's eq.?(1-x^2)y''-2xy'+a(a+1)y=0
Indicial Equation for Legendre's Eq. at x=1,-1
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- Thread starter sonia akram
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Discussion Overview
The discussion revolves around finding the indicial equation associated with the regular singularities at x=1 and x=-1 for Legendre's equation, specifically the equation (1-x^2)y''-2xy'+a(a+1)y=0. The scope includes theoretical exploration and mathematical reasoning related to differential equations.
Discussion Character
- Exploratory, Technical explanation, Mathematical reasoning
Main Points Raised
- One participant inquires about the indicial equation for the singularities at x=1 and x=-1.
- Another participant prompts the original poster to consider their own attempts at finding the solution, suggesting the use of a power series expansion.
- A different participant recalls that there may be a general formula for the indicial equation related to Legendre's equation, comparing it to the Euler-Cauchy differential equation.
- One participant clarifies that the series solution for singularity at x=0 differs from that for singularities at x=1 or x=-1, implying that the indicial equation would also be different.
- Another participant acknowledges the previous points and emphasizes the use of a series expansion centered around the singularity x=x0.
- A later reply expresses gratitude for the assistance provided in the discussion.
Areas of Agreement / Disagreement
The discussion reflects a lack of consensus on the specific form of the indicial equation for the singularities at x=1 and x=-1, with multiple viewpoints and approaches being presented.
Contextual Notes
Participants have not fully resolved the mathematical steps involved in deriving the indicial equation, and there are dependencies on definitions and assumptions regarding the series expansions used.
what is the indicial equation associated with regular singularities x=1 and x=-1 of legendre's eq.?(1-x^2)y''-2xy'+a(a+1)y=0
- #2
HallsofIvy
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What have you done to try to find it yourself?
If you write
[tex]y= \sum_{n=0}^\infty a_n x^{n+c}[/itex]<br /> what are y' and y"? What do you get when you put those into the equation?<br /> Assuming a_0 is not 0, what is the coefficient of the lowest power of x in that equation?[/tex]
If you write
[tex]y= \sum_{n=0}^\infty a_n x^{n+c}[/itex]<br /> what are y' and y"? What do you get when you put those into the equation?<br /> Assuming a_0 is not 0, what is the coefficient of the lowest power of x in that equation?[/tex]
- #3
matematikawan
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Don't they have a general formula for indicial equation for the Legendre's equation. I remember they have such a formula for Euler-Cauchy differential equation.
- #4
sonia akram
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HallsofIvy, that's the series solution for singularity x=0, if we are esprcially working for singularity x=1 or -1 than indicial would be different or not?
- #5
HallsofIvy
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Yes, I was using the most common application as an example. If a differential equation has a singularity at x= x0 you would use
[tex]y= \sum_{n=0}^\infty a_n (x- x_0)^{n+ c}[/tex]
[tex]y= \sum_{n=0}^\infty a_n (x- x_0)^{n+ c}[/tex]
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- #6
sonia akram
- 3
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thx a lot for ur help!
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