# Bessel equation

• June 13th 2008, 04:04 AM
panda
Bessel 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?
• June 13th 2008, 05:18 AM
CaptainBlack
Quote:

Originally Posted by panda
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?

Compute $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ in terms of $x$ and $\nu$ using the suggested $y(x)=x^{-1/2}\nu(x)$. It is labourious but it does give the quoted equation.

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

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

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

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

RonL