\[ \boxed { \left \{ {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) ={\frac {kx \left ( t \right ) }{ \left ( \left ( x \left ( t \right ) \right ) ^{2}+ \left ( y \left ( t \right ) \right ) ^{2} \right ) ^{3/2}}},{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}y \left ( t \right ) ={\frac {ky \left ( t \right ) }{ \left ( \left ( x \left ( t \right ) \right ) ^{2}+ \left ( y \left ( t \right ) \right ) ^{2} \right ) ^{3/2}}} \right \} } \]
Mathematica: cpu = 0.007001 (sec), leaf count = 52 \[ \text {DSolve}\left [\left \{x''(t)=\frac {k x(t)}{\left (x(t)^2+y(t)^2\right )^{3/2}},y''(t)=\frac {k y(t)}{\left (x(t)^2+y(t)^2\right )^{3/2}}\right \},\{x(t),y(t)\},t\right ] \]
Maple: cpu = 0 (sec), leaf count = 0 \[ \text {hanged} \]