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]