\[ \left \{x(t)=f\left (x'(t),y'(t)\right )+t x'(t),y(t)=g\left (x'(t),y'(t)\right )+t y'(t)\right \} \] ✗ Mathematica : cpu = 0.0043051 (sec), leaf count = 0 , could not solve
DSolve[{x[t] == f[Derivative[1][x][t], Derivative[1][y][t]] + t*Derivative[1][x][t], y[t] == g[Derivative[1][x][t], Derivative[1][y][t]] + t*Derivative[1][y][t]}, {x[t], y[t]}, t]
✓ Maple : cpu = 0.268 (sec), leaf count = 96
\[ \left \{ [ \left \{ \int \!{\it RootOf} \left ( g \left ( {\it \_Z},{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) -y \left ( t \right ) +t{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) \,{\rm d}t+{\it \_C1}=t{\it RootOf} \left ( g \left ( {\it \_Z},{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) -y \left ( t \right ) +t{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) +f \left ( {\it RootOf} \left ( g \left ( {\it \_Z},{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) -y \left ( t \right ) +t{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) ,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) \right \} , \left \{ x \left ( t \right ) =\int \!{\it RootOf} \left ( g \left ( {\it \_Z},{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) -y \left ( t \right ) +t{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) \,{\rm d}t+{\it \_C1} \right \} ] \right \} \]