While learning about Bessel functions I've came up with the following argument:
Since the Bessel equation is $$y''+\frac{1}{x} y'+ \left(1-\frac{\alpha^2}{x^2} \right) y=0, $$ one might expect that in the limit $x \to \infty$ we have $y''+y \approx 0$, which is the equation modelling simple harmonic motion (aka linear combinations of sines and cosines) of period $2 \pi$. Thus, we should have $y \approx C_1 \cos x+ C_2 \sin x$ for large $x$.
I'm aware that there is a branch of mathematics, called asymptotics (?) that deals with such ideas. Nonetheless I have several questions:
How can you know what terms are to be omitted as $x$ gets large? perhaps $y'$ grows faster than $\frac{1}{x}$ decays (so that the middle term cannot be omitted).
Looking at the graph of the Bessel function $J_0(x)$ one can see that the amplitude of the oscillation decays. Is there a way to predict that, and know the rate at which $C_1,C_2$ decay?
In general, is there a rigorous statement of the form: If $y' \approx G(x,y)$ is an approximation of the differential equation $y'=F(x,y)$ at the point $x_0$, with a solution $y=\varphi$, then the original equation has a solution $y$ which is $\approx \varphi$ near $x_0$? Hopefully with a precise notion of what "$\approx$" means.
Thank you!
I do not know how much this will help you; so, please, forgive me if I am off-topic.
The solution of the differential equation $$y''+\frac{1}{x} y'+ \left(1-\frac{\alpha^2}{x^2} \right) y=0$$ is given by $$y=c_1 J_{\alpha }(x)+c_2 Y_{\alpha }(x)$$ where appears the Bessel functions of the first and second kinds.
Considering the case where $x$ is large, the following first order approximations are available $$J_{\alpha }(x)\approx \sqrt{\frac{2}{\pi x }} \cos \left(\frac{\pi \alpha }{2}+\frac{\pi }{4}-x\right)$$ $$Y_{\alpha }(x)\approx -\sqrt{\frac{2}{\pi x }} \sin \left(\frac{\pi a}{2}+\frac{\pi }{4}-x\right)$$
Have a look here and here for more details about Hankel's expansions.