For a projectile on a horizontal plane, if given values for the angle of projection $\alpha$, the initial velocity $u$, some constant due to friction $k$ and an acceleration due to air resistance of: $$ a_x = k({v_x}^2 + {v_y}^2)\cos\beta $$ $$ a_y = -g - k({v_x}^2 + {v_y}^2)\sin\beta $$ where, at a given point $$ \beta = tan^{-1}\frac{v_y}{v_x} $$ Is it possible to calculate the usual parameters of time of flight, range, max height, landing velocity etc. by hand?
I am aware that one can solve for above by mapping a large series of positions over small increments of time using computer analysis, however I am intrigued to know if there is a more elegant solution than a brute-force algorithm.
EDIT: Are there any conditions with which this system would approximate a closed-form solution? E.g. making $\alpha$ small (and thus $v_y$) or changing the velocity in some way.