From Ottawa Citizen (and all over, really):

An Indian-born teenager has won a research award for solving a

mathematical problem first posed by Sir Isaac Newton more than 300

years ago that has baffled mathematicians ever since.The solution devised by Shouryya Ray, 16, makes it possible to

calculate exactly the path of a projectile under gravity and subject

to air resistance.This subject is of particular interest to me. I have been unable to locate his findings via the Internet. Where can I read his actual mathematical work?

Edit:

So has he written an actual paper, and if so, will anyone get to read it?

**Answer**

In the document *Comments on some recentwork by Shouryya Ray* by Prof. Dr. Ralph Chil and Prof. Dr. Jürgen Voigt (Technische Universität Dresden), dated June 4, 2012 it is written:

Conducting an internship at the Chair of Fluid Mechanics at TU

Dresden, Shouryya Ray encountered two ordinary differential equations

which are special cases of Newton’s law that the derivative of the

momentum of a particle equals the forces acting on it. In the first

one, which describes the motion of a particle in a gas or fluid, this

force is the sum of a damping force, which depends quadratically on

the velocity, and the (constant) gravitational force.

$$\begin{eqnarray*} \dot{u} &=&-u\sqrt{u^{2}+v^{2}},\qquad

u(0)=u_{0}>0 \\ \dot{v} &=&-v\sqrt{u^{2}+v^{2}}-g,\quad v(0)=v_{0}.

\end{eqnarray*}\tag{1}$$ Here, $u$ and $v$ are the horizontal and

vertical velocity, respectively.(…)

The second equation reads $$ \ddot{z}=-\dot{z}-z^{3/2},\qquad

z(0)=0,\dot{z}(0)=z_{1},\tag{2} $$ and describes the trajectory of the

center point $z(t)$ of a spherical particle during a normal collision

with a plane wall.

(…)Let us come back to problem (1) which was the starting point of the media stories. In the context of Shouryya Ray’s work it was an unfortunate circumstance, that a recent article from 2007$^8$ claims that no analytical solution of problem (1) was known, or that it was known only in special cases, namely falling objects$^9$. This might have misled Shouryya Ray who was not aware of the classical theory of ordinary differential equations.

(…)To conclude, Shouryya Ray has obtained analytic solutions of the problem (1), by transforming it successively to the problems (3)-(5), and by applying a recent result of D. Dominici in order to obtain a recursive formula for the coefficients of the power series representation of $\psi$. He then validated his results numerically. Given the level of prerequisites that he had, he made great progress. Nevertheless all his steps are basically known to experts and we emphasize that he did not solve an open problem posed by Newton.

(…)We hope that this small text gives the necessary information to the mathematical community, and that it allows the community to both put in context and appreciate the work of Shouryya Ray who plans to start a career in mathematics and physics.

The function $\psi$ is given by

$$\psi (t)=(v_{0}-g\Psi (t))/u_{0},$$

where

$$\Psi (t)=\int_{0}^{t}\exp \left[ \int_{0}^{\tau }\sqrt{u^{2}(s)+v^{2}(s)}ds

\right] d\tau .$$

I’ve read about this text on this blog post.

PS. Also in Spanish the Francis (th)E mule Science’s News post El problema de Newton y la solución que ha obtenido Shouryya Ray (16 años) discusses these problems.

**Attribution***Source : Link , Question Author : jnm2 , Answer Author : Américo Tavares*