On integrating the projectile motion equations of a heavy point in medium with height decreasing density

The resolvent method based on Legendre transformation was applied to integrate ballistic equations of a heavy point mass in inhomogeneous medium with the drag force being power-law with respect to speed, at that the coefficient of the drag force decreases linearly with height $y$. General expressions were obtained for resolvent function $a''_{bb}(b)$ with $a(b)$ being an intercept and $b=\operatorname{tg}\theta$, where $\theta$ is inclination angle. In the second order by gradient $c/m^{-1}$ of perturbative approach, the universal formulas for $\delta a''_{bb}(b)$-, $\delta x(b)$-, $\delta y(b)$-additions were derived. The case of Releigh resistance was considered particularly in frames of the method above and inhomogeneity influence on the motion was investigated. The falling of gravity $g(y)$ is taken into consideration too.