the question says 'use the result to solve bessel's equation of index n=1/2'.

i wasn't sure what it means by index n=1/2?

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- Jun 13th 2008, 04:04 AMpandaBessel equation
the question says 'use the result to solve bessel's equation of index n=1/2'.

i wasn't sure what it means by index n=1/2? - Jun 13th 2008, 05:18 AMCaptainBlack
Compute $\displaystyle \frac{dy}{dx}$ and $\displaystyle \frac{d^2y}{dx^2}$ in terms of $\displaystyle x$ and $\displaystyle \nu$ using the suggested $\displaystyle y(x)=x^{-1/2}\nu(x)$. It is labourious but it does give the quoted equation.

Now if we put $\displaystyle n=1/2$ in the final equation it reduces to:

$\displaystyle

\frac{d^2\nu}{dx^2}+\nu=0

$

which has general solution $\displaystyle \nu(x)=A\cos(x)+B\sin(x)$, so the solution of the Bessel equation with $\displaystyle n=1/2$ (that is of order $\displaystyle n=1/2$) is:

$\displaystyle

y(x)=x^{-1/2}[A\cos(x)+B\sin(x)]

$

RonL