\[ \boxed { \left \{ \left ( \left ( x \left ( t \right ) \right ) ^{2}+ \left ( y \left ( t \right ) \right ) ^{2}-{t}^{2} \right ) {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) =-2\,tx \left ( t \right ) , \left ( \left ( x \left ( t \right ) \right ) ^{2}+ \left ( y \left ( t \right ) \right ) ^{2}-{t}^{2} \right ) {\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =-2\,ty \left ( t \right ) \right \} } \]
Mathematica: cpu = 0.071009 (sec), leaf count = 191 \[ \left \{\left \{y(t)\to \frac {c_1 \left (e^{c_2}-\sqrt {-4 c_1^2 t^2+e^{2 c_2}-4 t^2}\right )}{2 \left (c_1^2+1\right )},x(t)\to \frac {e^{c_2}-\sqrt {-4 c_1^2 t^2+e^{2 c_2}-4 t^2}}{2 \left (c_1^2+1\right )}\right \},\left \{y(t)\to \frac {c_1 \left (\sqrt {-4 c_1^2 t^2+e^{2 c_2}-4 t^2}+e^{c_2}\right )}{2 \left (c_1^2+1\right )},x(t)\to \frac {\sqrt {-4 c_1^2 t^2+e^{2 c_2}-4 t^2}+e^{c_2}}{2 \left (c_1^2+1\right )}\right \}\right \} \]
Maple: cpu = 0.234 (sec), leaf count = 186 \[ \left \{ [ \left \{ x \left ( t \right ) =0 \right \} , \left \{ y \left ( t \right ) ={\frac {1}{2\,{\it \_C1}} \left ( 1+\sqrt {-4\,{{\it \_C1}}^{ 2}{t}^{2}+1} \right ) },y \left ( t \right ) =-{\frac {1}{2\,{\it \_C1}} \left ( -1+\sqrt {-4\,{{\it \_C1}}^{2}{t}^{2}+1} \right ) } \right \} ], [ \left \{ x \left ( t \right ) =-{\frac {1}{2\,{\it \_C2}} \left ( -{\it \_C1}+\sqrt {-2\,{\it \_C2}\,{t}^{2}+{{\it \_C1}}^{2}} \right ) },x \left ( t \right ) ={\frac {1}{2\,{\it \_C2}} \left ( {\it \_C1}+\sqrt { -2\,{\it \_C2}\,{t}^{2}+{{\it \_C1}}^{2}} \right ) } \right \} , \left \{ y \left ( t \right ) ={\frac {1}{{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) }\sqrt {- \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) \left ( \left ( x \left ( t \right ) \right ) ^{2}{ \frac {\rm d}{{\rm d}t}}x \left ( t \right ) - \left ( {\frac {\rm d}{ {\rm d}t}}x \left ( t \right ) \right ) {t}^{2}+2\,tx \left ( t \right ) \right ) }},y \left ( t \right ) =-{\frac {1}{{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) }\sqrt {- \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) \left ( \left ( x \left ( t \right ) \right ) ^{2}{ \frac {\rm d}{{\rm d}t}}x \left ( t \right ) - \left ( {\frac {\rm d}{ {\rm d}t}}x \left ( t \right ) \right ) {t}^{2}+2\,tx \left ( t \right ) \right ) }} \right \} ] \right \} \]