\[ \left \{x''(t)=-\frac {c y(t) x'(t) f\left (\sqrt {x'(t)^2+y'(t)^2}\right )}{\sqrt {x'(t)^2+y'(t)^2}},y''(t)=-\frac {c y(t) y'(t) f\left (\sqrt {x'(t)^2+y'(t)^2}\right )}{\sqrt {x'(t)^2+y'(t)^2}}-g\right \} \] ✗ Mathematica : cpu = 0.0072309 (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 = 4.415 (sec), leaf count = 116
\[\left \{\left [\left \{y \left (t \right ) = \mathit {ODESolStruc} \left (\textit {\_a} , \left [\left \{\textit {\_}b\left (\textit {\_a} \right ) \left (\frac {d}{d \textit {\_a}}\mathrm {\_}\mathrm {b}\left (\textit {\_a} \right )\right )+\frac {C \left (\textit {\_a} \right ) \textit {\_}b\left (\textit {\_a} \right ) f \left (\sqrt {\textit {\_}b\left (\textit {\_a} \right )^{2}}\right )+\sqrt {\textit {\_}b\left (\textit {\_a} \right )^{2}}\, g}{\sqrt {\textit {\_}b\left (\textit {\_a} \right )^{2}}}=0\right \}, \left \{\textit {\_a} =y \left (t \right ), \textit {\_}b\left (\textit {\_a} \right )=\frac {d}{d t}y \left (t \right )\right \}, \left \{t =c_{3}+\int \frac {1}{\textit {\_}b\left (\textit {\_a} \right )}d \textit {\_a} , y \left (t \right )=\textit {\_a} \right \}\right ]\right )\right \}, \left \{x \left (t \right ) = c_{2} \left (\int {\mathrm e}^{\int -\frac {\sqrt {\left (\frac {d}{d t}y \left (t \right )\right )^{2}}\, C \left (y \left (t \right )\right ) f \left (\sqrt {\left (\frac {d}{d t}y \left (t \right )\right )^{2}}\right )}{\left (\frac {d}{d t}y \left (t \right )\right )^{2}}d t}d t \right )+c_{1}\right \}\right ]\right \}\]