\[ \left \{a y'(t)+t x'(t)-x(t)+y'(t)^2=0,x'(t) y'(t)+t y'(t)-y(t)=0\right \} \] ✗ Mathematica : cpu = 8.72384 (sec), leaf count = 0 , could not solve
DSolve[{-x[t] + t*Derivative[1][x][t] + a*Derivative[1][y][t] + Derivative[1][y][t]^2 == 0, -y[t] + t*Derivative[1][y][t] + Derivative[1][x][t]*Derivative[1][y][t] == 0}, {x[t], y[t]}, t]
✓ Maple : cpu = 0.312 (sec), leaf count = 194
\[ \left \{ [ \left \{ x \left ( t \right ) =-{\frac {{t}^{2}}{3}} \right \} , \left \{ y \left ( t \right ) =-{\frac {{t}^{3}}{27\,a}} \right \} ],[ \left \{ x \left ( t \right ) ={\it \_C1}\,t+{\it \_C2} \right \} , \left \{ y \left ( t \right ) =-{\frac { \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +t \right ) \left ( \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{2}+t{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) -x \left ( t \right ) \right ) }{a}} \right \} ],[ \left \{ x \left ( t \right ) ={\frac {-3\,{{\it \_C1}}^{2}{t}^{2}-2\,\sqrt {3}{\it \_C1}\,t+3}{12\,{{\it \_C1}}^{2}}},x \left ( t \right ) ={\frac {-3\,{{\it \_C1}}^{2}{t}^{2}+2\,\sqrt {3}{\it \_C1}\,t+3}{12\,{{\it \_C1}}^{2}}},x \left ( t \right ) =-{\frac {\sqrt {3}{\it \_C1}\,t}{6}}+{\frac {{{\it \_C1}}^{2}}{4}}-{\frac {{t}^{2}}{4}},x \left ( t \right ) ={\frac {\sqrt {3}{\it \_C1}\,t}{6}}+{\frac {{{\it \_C1}}^{2}}{4}}-{\frac {{t}^{2}}{4}} \right \} , \left \{ y \left ( t \right ) ={\frac { \left ( 2\,{t}^{2}+6\,x \left ( t \right ) \right ) {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +2\,{t}^{3}+7\,tx \left ( t \right ) }{9\,a}} \right \} ] \right \} \]