\[ \left \{x''(t)=-\frac {c y(t) x'(t) f\left (\sqrt {x'(t)^2+y'(t)^2}\right )}{\sqrt {x'(t)^2+y'(t)^2}},y''(t)=-\frac {c y(t) y'(t) f\left (\sqrt {x'(t)^2+y'(t)^2}\right )}{\sqrt {x'(t)^2+y'(t)^2}}-g\right \} \] ✗ Mathematica : cpu = 0.0064623 (sec), leaf count = 0 , could not solve
DSolve[{Derivative[2][x][t] == -((c*f[Sqrt[Derivative[1][x][t]^2 + Derivative[1][y][t]^2]]*y[t]*Derivative[1][x][t])/Sqrt[Derivative[1][x][t]^2 + Derivative[1][y][t]^2]), Derivative[2][y][t] == -g - (c*f[Sqrt[Derivative[1][x][t]^2 + Derivative[1][y][t]^2]]*y[t]*Derivative[1][y][t])/Sqrt[Derivative[1][x][t]^2 + Derivative[1][y][t]^2]}, {x[t], y[t]}, t]
✓ Maple : cpu = 11.667 (sec), leaf count = 116
\[ \left \{ [ \left \{ y \left ( t \right ) ={\it ODESolStruc} \left ( {\it \_a},[ \left \{ \left ( {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) \right ) {\it \_b} \left ( {\it \_a} \right ) +{ \left ( C \left ( {\it \_a} \right ) f \left ( \sqrt { \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}} \right ) {\it \_b} \left ( {\it \_a} \right ) +g\sqrt { \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}} \right ) {\frac {1}{\sqrt { \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}}}}}=0 \right \} , \left \{ {\it \_a}=y \left ( t \right ) ,{\it \_b} \left ( {\it \_a} \right ) ={\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right \} , \left \{ t=\int \! \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{-1}\,{\rm d}{\it \_a}+{\it \_C3},y \left ( t \right ) ={\it \_a} \right \} ] \right ) \right \} , \left \{ x \left ( t \right ) ={\it \_C1}+\int \!{{\rm e}^{\int \!-{\frac {C \left ( y \left ( t \right ) \right ) }{ \left ( {\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) ^{2}}\sqrt { \left ( {\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) ^{2}}f \left ( \sqrt { \left ( {\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) ^{2}} \right ) }\,{\rm d}t}}\,{\rm d}t{\it \_C2} \right \} ] \right \} \]