Some asymptotics for the Bessel functions with an explicit error term

We show how one can obtain an asymptotic expression for some special functions satisfying a second order differential equation with a very explicit error term starting from appropriate upper bounds. We will work out the details for the Bessel function $J_\nu (x)$ and the Airy function $Ai(x)$ and find a sharp approximation for their zeros. We also answer the question raised by Olenko by showing that $$c_1 | \nu^2-1/4\,| < \sup_{x \ge 0} x^{3/2}|J_\nu(x)-\sqrt{\frac{2}{\pi x}} \, \cos (x-\frac{\pi \nu}{2}-\frac{\pi}{4}\,)| <c_2 |\nu^2-1/4\,|, $$ $ \nu \ge -1/2 \, ,$ for some explicit numerical constants $c_1$ and $c_2.$