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Transforming an ODE into Legendre's Equation

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1

$begingroup$

I am trying to transform the ODE $$frac1sin(theta)fracddthetaleft(sin(theta)fracdSdthetaright)+lambda S=0,$$ in to Legendre's equation $$(1-mu^2)fracd^2Sdmu^2-2mufracdSdmu+n(n+1)S=0$$ when $lambda=n(n+1)$ for $n=0,1,2..$ and $mu=cos(theta)$.

I calculated that,
beginalign
fracdSdtheta&=fracdSdmufracdmudtheta=-sin(theta)fracdSdmu \
fracd^2Sdtheta^2&=fracddthetaleft(fracdSdthetaright)=-cos(theta)fracd^2Sdmu^2.
endalign
Then,
beginalign
frac1sin(theta)left(sin(theta)fracd^2Sdtheta^2+cos(theta)fracdSdthetaright)+Slambda&=0 \
-mufracd^2Sdmu^2-mufracdSdmu+n(n+1)S&=0.
endalign
But at this point, I don't see how Legendre's equation is possible. I know that the double derivative is calculated incorrectly, but I don't know how to rectify this.

ordinary-differential-equations derivatives legendre-functions

share|cite|improve this question

edited Mar 15 at 3:41

Bell

asked Mar 15 at 3:30

BellBell

196318

$endgroup$

add a comment | 

1

$begingroup$

I am trying to transform the ODE $$frac1sin(theta)fracddthetaleft(sin(theta)fracdSdthetaright)+lambda S=0,$$ in to Legendre's equation $$(1-mu^2)fracd^2Sdmu^2-2mufracdSdmu+n(n+1)S=0$$ when $lambda=n(n+1)$ for $n=0,1,2..$ and $mu=cos(theta)$.

I calculated that,
beginalign
fracdSdtheta&=fracdSdmufracdmudtheta=-sin(theta)fracdSdmu \
fracd^2Sdtheta^2&=fracddthetaleft(fracdSdthetaright)=-cos(theta)fracd^2Sdmu^2.
endalign
Then,
beginalign
frac1sin(theta)left(sin(theta)fracd^2Sdtheta^2+cos(theta)fracdSdthetaright)+Slambda&=0 \
-mufracd^2Sdmu^2-mufracdSdmu+n(n+1)S&=0.
endalign
But at this point, I don't see how Legendre's equation is possible. I know that the double derivative is calculated incorrectly, but I don't know how to rectify this.

ordinary-differential-equations derivatives legendre-functions

share|cite|improve this question

edited Mar 15 at 3:41

Bell

asked Mar 15 at 3:30

BellBell

196318

$endgroup$

add a comment | 

$begingroup$

I am trying to transform the ODE $$frac1sin(theta)fracddthetaleft(sin(theta)fracdSdthetaright)+lambda S=0,$$ in to Legendre's equation $$(1-mu^2)fracd^2Sdmu^2-2mufracdSdmu+n(n+1)S=0$$ when $lambda=n(n+1)$ for $n=0,1,2..$ and $mu=cos(theta)$.

I calculated that,
beginalign
fracdSdtheta&=fracdSdmufracdmudtheta=-sin(theta)fracdSdmu \
fracd^2Sdtheta^2&=fracddthetaleft(fracdSdthetaright)=-cos(theta)fracd^2Sdmu^2.
endalign
Then,
beginalign
frac1sin(theta)left(sin(theta)fracd^2Sdtheta^2+cos(theta)fracdSdthetaright)+Slambda&=0 \
-mufracd^2Sdmu^2-mufracdSdmu+n(n+1)S&=0.
endalign
But at this point, I don't see how Legendre's equation is possible. I know that the double derivative is calculated incorrectly, but I don't know how to rectify this.

ordinary-differential-equations derivatives legendre-functions

share|cite|improve this question

edited Mar 15 at 3:41

Bell

asked Mar 15 at 3:30

BellBell

196318

$endgroup$

share|cite|improve this question

edited Mar 15 at 3:41

Bell

asked Mar 15 at 3:30

BellBell

196318

share|cite|improve this question

edited Mar 15 at 3:41

Bell

asked Mar 15 at 3:30

BellBell

196318

asked Mar 15 at 3:30

BellBell

196318

asked Mar 15 at 3:30

BellBell

196318