- #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
- Thread starter sonia akram
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In summary, the indicial equation associated with the regular singularities x=1 and x=-1 of Legendre's equation is given by y= \sum_{n=0}^\infty a_n (x- x_0)^{n+ c}, where y' and y" are the derivatives of y, and the coefficient of the lowest power of x in the equation is used to find the general formula for the indicial equation for Euler-Cauchy differential equations. This method can be applied to any singularity at x=x0.
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]
what are y' and y"? What do you get when you put those into the equation?
Assuming a_0 is not 0, what is the coefficient of the lowest power of x in that equation?
If you write
[tex]y= \sum_{n=0}^\infty a_n x^{n+c}[/itex]
what are y' and y"? What do you get when you put those into the equation?
Assuming a_0 is not 0, what is the coefficient of the lowest power of x in that equation?
- #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
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thx a lot for ur help!
Related to Indicial Equation for Legendre's Eq. at x=1,-1
What is the Indicial Equation for Legendre's Equation at x=1?
The Indicial Equation for Legendre's Equation at x=1 is given by the equation:
m(m+1) = l(l+1), where m is the index and l is the degree of the Legendre polynomial.
What is the significance of the Indicial Equation for Legendre's Equation?
The Indicial Equation helps determine the type of solutions for Legendre's Equation.
If the roots of the Indicial Equation are real and distinct, the solutions are in the form of Legendre polynomials.
If the roots are the same, the solutions are in the form of logarithmic functions.
And if the roots are complex, the solutions are in the form of modified Bessel functions.
How do you solve the Indicial Equation for Legendre's Equation at x=1?
To solve the Indicial Equation at x=1, we substitute x=1 into the equation and solve for the index m.
Then, the degree l can be determined by using the formula: l = m+1 or l = -m.
The values of m and l will determine the type of solutions for Legendre's Equation.
What is the Indicial Equation for Legendre's Equation at x=-1?
The Indicial Equation for Legendre's Equation at x=-1 is given by the equation:
m(m+1) = (-1)^l * l(l+1), where m is the index and l is the degree of the Legendre polynomial.
This equation takes into account the alternating sign of Legendre polynomials at x=-1.
What are the possible solutions for Legendre's Equation based on the Indicial Equation at x=1,-1?
The possible solutions for Legendre's Equation are Legendre polynomials, logarithmic functions, and modified Bessel functions.
The specific type of solution depends on the values of m and l, which are determined by the roots of the Indicial Equation at x=1,-1.
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