\[ \boxed { \left \{ \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) {\frac {\rm d}{{\rm d}t}}y \left ( t \right ) +t{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) -y \left ( t \right ) =0, \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{2}+t{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +a{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) -x \left ( t \right ) =0 \right \} } \]
Mathematica: cpu = 8.399567 (sec), leaf count = 55 \[ \text {DSolve}\left [\left \{a y'(t)+t x'(t)-x(t)+y'(t)^2=0,x'(t) y'(t)+t y'(t)-y(t)=0\right \},\{x(t),y(t)\},t\right ] \]
Maple: cpu = 0.219 (sec), leaf count = 226 \[ \left \{ [ \left \{ x \left ( t \right ) =-{\frac {{t}^{2}}{3}} \right \} , \left \{ y \left ( t \right ) =-{\frac {{t}^{3}}{27\,a}} \right \} ],[ \left \{ x \left ( t \right ) ={\it \_C1}\,t+{\it \_C2} \right \} , \left \{ y \left ( t \right ) ={\frac {- \left ( {\frac {\rm d}{{\rm d}t} }x \left ( t \right ) \right ) ^{3}-2\, \left ( {\frac {\rm d}{{\rm d}t}} x \left ( t \right ) \right ) ^{2}t- \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) {t}^{2}+x \left ( t \right ) {\frac {\rm d}{ {\rm d}t}}x \left ( t \right ) +tx \left ( t \right ) }{a}} \right \} ],[ \left \{ x \left ( t \right ) =-{\frac {5\,{{\it \_C1}}^{2}{t}^{2}+2\,{ \it \_C1}\,t \left ( -{\it \_C1}\,t+\sqrt {3} \right ) -3}{12\,{{\it \_C1}}^{2}}},x \left ( t \right ) =-{\frac {5\,{{\it \_C1}}^{2}{t}^{2}-2 \,{\it \_C1}\,t \left ( {\it \_C1}\,t+\sqrt {3} \right ) -3}{12\,{{\it \_C1}}^{2}}},x \left ( t \right ) =-{\frac {5\,{t}^{2}}{12}}+{\frac {t \left ( t-\sqrt {3}{\it \_C1} \right ) }{6}}+{\frac {{{\it \_C1}}^{2}}{ 4}},x \left ( t \right ) =-{\frac {5\,{t}^{2}}{12}}+{\frac {t \left ( t+ \sqrt {3}{\it \_C1} \right ) }{6}}+{\frac {{{\it \_C1}}^{2}}{4}} \right \} , \left \{ y \left ( t \right ) =-{\frac {-2\, \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) {t}^{2}-2\,{t}^{3}-6\,x \left ( t \right ) {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) -7\,tx \left ( t \right ) }{9\,a}} \right \} ] \right \} \]