\[ \boxed { \left \{ x \left ( t \right ) =t{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +f \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) ,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) ,y \left ( t \right ) =t{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) +g \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) ,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) \right \} } \]
Mathematica: cpu = 0.005501 (sec), leaf count = 46 \[ \text {DSolve}\left [\left \{x(t)=f\left (x'(t),y'(t)\right )+t x'(t),y(t)=g\left (x'(t),y'(t)\right )+t y'(t)\right \},\{x(t),y(t)\},t\right ] \]
Maple: cpu = 0.078 (sec), leaf count = 96 \[ \left \{ [ \left \{ \int \!{\it RootOf} \left ( t{\frac {\rm d}{{\rm d}t }}y \left ( t \right ) +g \left ( {\it \_Z},{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) -y \left ( t \right ) \right ) \,{\rm d}t+{ \it \_C1}=t{\it RootOf} \left ( t{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) +g \left ( {\it \_Z},{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) -y \left ( t \right ) \right ) +f \left ( {\it RootOf} \left ( t{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) +g \left ( {\it \_Z},{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) -y \left ( t \right ) \right ) ,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) \right \} , \left \{ x \left ( t \right ) =\int \!{\it RootOf} \left ( t{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) +g \left ( {\it \_Z},{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) -y \left ( t \right ) \right ) \,{\rm d}t+{\it \_C1} \right \} ] \right \} \]