\[ \left \{x''(t)=-\frac {c y(t) x'(t) f\left (\sqrt {x'(t)^2+y'(t)^2}\right )}{\sqrt {x'(t)^2+y'(t)^2}},y''(t)=-\frac {c y(t) y'(t) f\left (\sqrt {x'(t)^2+y'(t)^2}\right )}{\sqrt {x'(t)^2+y'(t)^2}}-g\right \} \] ✗ Mathematica : cpu = 0.0068803 (sec), leaf count = 0
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]
, 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 = 0. (sec), leaf count = 0
dsolve({diff(diff(x(t),t),t) = -C(y(t))*f((diff(y(t),t)^2)^(1/2))/(diff(y(t),t)^2)^(1/2)*diff(x(t),t), diff(diff(y(t),t),t) = -C(y(t))*f((diff(y(t),t)^2)^(1/2))/(diff(y(t),t)^2)^(1/2)*diff(y(t),t)-g})
, result contains DESol or ODESolStruc
\[\left [\left \{y \left (t \right ) = \textit {\_a} \boldsymbol {\mathrm {where}}\left [\left \{\left (\frac {d}{d \textit {\_a}}\mathrm {\_}\mathrm {b}\left (\textit {\_a} \right )\right ) \textit {\_}b\left (\textit {\_a} \right )+\frac {C \left (\textit {\_a} \right ) f \left (\sqrt {\textit {\_}b\left (\textit {\_a} \right )^{2}}\right ) \textit {\_}b\left (\textit {\_a} \right )+g \sqrt {\textit {\_}b\left (\textit {\_a} \right )^{2}}}{\sqrt {\textit {\_}b\left (\textit {\_a} \right )^{2}}}=0\right \}, \left \{\textit {\_a} =y \left (t \right ), \textit {\_}b\left (\textit {\_a} \right )=\frac {d}{d t}y \left (t \right )\right \}, \left \{t =\int \frac {1}{\textit {\_}b\left (\textit {\_a} \right )}d \textit {\_a} +c_{3}, y \left (t \right )=\textit {\_a} \right \}\right ]\right \}, \left \{x \left (t \right ) = c_{1}+\left (\int {\mathrm e}^{\int -\frac {\sqrt {\left (\frac {d}{d t}y \left (t \right )\right )^{2}}\, C \left (y \left (t \right )\right ) f \left (\sqrt {\left (\frac {d}{d t}y \left (t \right )\right )^{2}}\right )}{\left (\frac {d}{d t}y \left (t \right )\right )^{2}}d t}d t \right ) c_{2}\right \}\right ]\]