Legendre's Equation, sturm liouville - eigenvalues/eigenfunction
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Linear Differential Equation,Legendre's Equation, sturm liouville - eigenvalues/eigenfunction
Consider the linear differential operator:
$$ L = frac{1}{4}(1+x^2)frac{d^2}{dx^2}+frac{1}{2}x(1+x^2)frac{d}{dx}+a $$
acting on functions defined in $-1 le x le 1$ and vanishing at the endpoints of the interval.
(a) Is $L$ Hermitian?
(b) Determine the weight function necessary to make $L$ Hermitian.
(c) Show explicitly that
$$
int_{-1}^{1}V^*(x)W(x)Lu(x)dx = int_{-1}^{1}(LV)^*W(x)u(x)dx
$$
and thereby determine the condition on 'a'.
(d) Change variables to
$$ x= tanleft(frac{Theta}{2}right) $$
Find $2$ even eigenfunctions $f_1(x)$ and $f_2(x)$ of the diferential equation
$$ Lu=lambda u. $$
It's my first time posting question, so wasn't sure how to type the differential equation.
eigenfunctions legendre-polynomials sturm-liouville
edited Nov 25 at 20:31
DisintegratingByParts
58.1k42477
asked Nov 25 at 14:19
cisko
61
add a comment |
up vote
-3
down vote
favorite
Linear Differential Equation,Legendre's Equation, sturm liouville - eigenvalues/eigenfunction
Consider the linear differential operator:
$$ L = frac{1}{4}(1+x^2)frac{d^2}{dx^2}+frac{1}{2}x(1+x^2)frac{d}{dx}+a $$
acting on functions defined in $-1 le x le 1$ and vanishing at the endpoints of the interval.
(a) Is $L$ Hermitian?
(b) Determine the weight function necessary to make $L$ Hermitian.
(c) Show explicitly that
$$
int_{-1}^{1}V^*(x)W(x)Lu(x)dx = int_{-1}^{1}(LV)^*W(x)u(x)dx
$$
and thereby determine the condition on 'a'.
(d) Change variables to
$$ x= tanleft(frac{Theta}{2}right) $$
Find $2$ even eigenfunctions $f_1(x)$ and $f_2(x)$ of the diferential equation
$$ Lu=lambda u. $$
It's my first time posting question, so wasn't sure how to type the differential equation.
eigenfunctions legendre-polynomials sturm-liouville
edited Nov 25 at 20:31
DisintegratingByParts
58.1k42477
asked Nov 25 at 14:19
cisko
61
Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
– José Carlos Santos
Nov 25 at 14:21
I know, I am new to this and still practicing. The equations weren't coming the way I wanted them to.
– cisko
Nov 25 at 15:11
You can take a look at how I edited your question. That will help you learn MathJax for the next time.
– DisintegratingByParts
Nov 25 at 20:33
Thank you so much
– cisko
Nov 26 at 21:02
add a comment |
up vote
-3
down vote
favorite
up vote
-3
down vote
favorite
Linear Differential Equation,Legendre's Equation, sturm liouville - eigenvalues/eigenfunction
Consider the linear differential operator:
$$ L = frac{1}{4}(1+x^2)frac{d^2}{dx^2}+frac{1}{2}x(1+x^2)frac{d}{dx}+a $$
acting on functions defined in $-1 le x le 1$ and vanishing at the endpoints of the interval.
(a) Is $L$ Hermitian?
(b) Determine the weight function necessary to make $L$ Hermitian.
(c) Show explicitly that
$$
int_{-1}^{1}V^*(x)W(x)Lu(x)dx = int_{-1}^{1}(LV)^*W(x)u(x)dx
$$
and thereby determine the condition on 'a'.
(d) Change variables to
$$ x= tanleft(frac{Theta}{2}right) $$
Find $2$ even eigenfunctions $f_1(x)$ and $f_2(x)$ of the diferential equation
$$ Lu=lambda u. $$
It's my first time posting question, so wasn't sure how to type the differential equation.
eigenfunctions legendre-polynomials sturm-liouville
edited Nov 25 at 20:31
DisintegratingByParts
58.1k42477
asked Nov 25 at 14:19
cisko
61
Linear Differential Equation,Legendre's Equation, sturm liouville - eigenvalues/eigenfunction
Consider the linear differential operator:
$$ L = frac{1}{4}(1+x^2)frac{d^2}{dx^2}+frac{1}{2}x(1+x^2)frac{d}{dx}+a $$
acting on functions defined in $-1 le x le 1$ and vanishing at the endpoints of the interval.
(a) Is $L$ Hermitian?
(b) Determine the weight function necessary to make $L$ Hermitian.
(c) Show explicitly that
$$
int_{-1}^{1}V^*(x)W(x)Lu(x)dx = int_{-1}^{1}(LV)^*W(x)u(x)dx
$$
and thereby determine the condition on 'a'.
(d) Change variables to
$$ x= tanleft(frac{Theta}{2}right) $$
Find $2$ even eigenfunctions $f_1(x)$ and $f_2(x)$ of the diferential equation
$$ Lu=lambda u. $$
It's my first time posting question, so wasn't sure how to type the differential equation.
eigenfunctions legendre-polynomials sturm-liouville
eigenfunctions legendre-polynomials sturm-liouville
edited Nov 25 at 20:31
DisintegratingByParts
58.1k42477
asked Nov 25 at 14:19
cisko
61
edited Nov 25 at 20:31
DisintegratingByParts
58.1k42477
asked Nov 25 at 14:19
cisko
61
edited Nov 25 at 20:31
DisintegratingByParts
58.1k42477
edited Nov 25 at 20:31
DisintegratingByParts
58.1k42477
edited Nov 25 at 20:31
DisintegratingByParts
58.1k42477
58.1k42477
asked Nov 25 at 14:19
cisko
61
asked Nov 25 at 14:19
cisko
61
asked Nov 25 at 14:19
cisko
61
61
Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
– José Carlos Santos
Nov 25 at 14:21
I know, I am new to this and still practicing. The equations weren't coming the way I wanted them to.
– cisko
Nov 25 at 15:11
You can take a look at how I edited your question. That will help you learn MathJax for the next time.
– DisintegratingByParts
Nov 25 at 20:33
Thank you so much
– cisko
Nov 26 at 21:02
add a comment |
Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
– José Carlos Santos
Nov 25 at 14:21
I know, I am new to this and still practicing. The equations weren't coming the way I wanted them to.
– cisko
Nov 25 at 15:11
You can take a look at how I edited your question. That will help you learn MathJax for the next time.
– DisintegratingByParts
Nov 25 at 20:33
Thank you so much
– cisko
Nov 26 at 21:02
Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
– José Carlos Santos
Nov 25 at 14:21
Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
– José Carlos Santos
Nov 25 at 14:21
I know, I am new to this and still practicing. The equations weren't coming the way I wanted them to.
– cisko
Nov 25 at 15:11
I know, I am new to this and still practicing. The equations weren't coming the way I wanted them to.
– cisko
Nov 25 at 15:11
You can take a look at how I edited your question. That will help you learn MathJax for the next time.
– DisintegratingByParts
Nov 25 at 20:33
You can take a look at how I edited your question. That will help you learn MathJax for the next time.
– DisintegratingByParts
Nov 25 at 20:33
Thank you so much
– cisko
Nov 26 at 21:02
Thank you so much
– cisko
Nov 26 at 21:02
add a comment |
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Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
– José Carlos Santos
Nov 25 at 14:21
I know, I am new to this and still practicing. The equations weren't coming the way I wanted them to.
– cisko
Nov 25 at 15:11
You can take a look at how I edited your question. That will help you learn MathJax for the next time.
– DisintegratingByParts
Nov 25 at 20:33
Thank you so much
– cisko
Nov 26 at 21:02