Bessel ODE 3
The ordinary differential equation by Bessel is defined by:
$$
x^2\, y_\lambda'' + x\, y_\lambda' + (x^2 - \lambda^2)\, y_\lambda = 0
$$
In the previous chapter
we have found ladder relations for solutions of the Bessel equation:
$$
y_{\lambda-1}(x)=\left[\frac{d}{dx}+\frac{\lambda}{x}\right]y_{\lambda}(x) \\
y_{\lambda+1}(x)=\left[\frac{d}{dx}-\frac{\lambda}{x}\right]y_{\lambda}(x)
$$
Using again (and again) the formula at the bottom line of our
general theory, the operators of the ladder
relations can be rewritten as:
$$
\frac{d}{dx}\pm\frac{\lambda}{x}=
e^{\mp\int\lambda/x\,dx}\frac{d}{dx}e^{\pm\int\lambda/x\,dx}=
e^{\mp\lambda\ln(x)}\frac{d}{dx}e^{\pm\lambda\ln(x)}=
x^{\mp\lambda}\frac{d}{dx}x^{\pm\lambda}
$$
From:
$$
y_{\lambda-1} = x^{-\lambda}\frac{d}{dx}x^{+\lambda} y_\lambda
\qquad \mbox{and} \qquad
y_{\lambda+1} = x^{+\lambda}\frac{d}{dx}x^{-\lambda} y_\lambda
$$ it follows that: $$
y_{\lambda-2} = \left(x^{-(\lambda-1)}\frac{d}{dx}x^{+(\lambda-1)}\right)
\left(x^{-\lambda}\frac{d}{dx}x^{+\lambda}\right) y_\lambda =
x^{-(\lambda-2)}\frac{1}{x}\frac{d}{dx}\frac{1}{x}\frac{d}{dx}x^{\lambda} y_\lambda =
x^{-(\lambda-2)}\left(\frac{1}{x}\frac{d}{dx}\right)^2 x^{\lambda} y_\lambda \\
y_{\lambda+2} = \left(x^{+(\lambda+1)}\frac{d}{dx}x^{-(\lambda+1)}\right)
\left(x^{+\lambda}\frac{d}{dx}x^{-\lambda}\right) y_\lambda =
x^{+(\lambda+2)}\frac{1}{x}\frac{d}{dx}\frac{1}{x}\frac{d}{dx}x^{\lambda} y_\lambda =
x^{+(\lambda+2)}\left(\frac{1}{x}\frac{d}{dx}\right)^2 x^{\lambda} y_\lambda
$$
It is conjectured that for $n\in\mathbb{N}$ (proof by induction eventually):
$$
y_{\lambda-n}(x) =
x^{-(\lambda-n)}\left(\frac{1}{x}\frac{d}{dx}\right)^n x^{\lambda}\, y_\lambda(x) \\
y_{\lambda+n}(x) =
x^{+(\lambda+n)}\left(\frac{1}{x}\frac{d}{dx}\right)^n x^{\lambda}\, y_\lambda(x)
$$
When applied to the solutions that have been found in another
previous chapter, namely:
$$
y_{1/2}(x) = \frac{C_1 \cos(x) + C_2 \sin(x)}{\sqrt{x}}
$$
Then we have found all solutions of the Bessel equation
for $n\in\mathbb{N}$ in $\lambda = n + 1/2$ :
$$
y_{n+1/2} = y_{-n-1/2} =
x^{n+1/2}\left(\frac{1}{x}\frac{d}{dx}\right)^n \frac{C_1 \cos(x) + C_2 \sin(x)}{x}
$$