\[ \boxed { \begin {array}{rl} x''(t) &= \frac {\partial F}{\partial x}\\ y''(t) &= \frac {\partial F}{\partial y}\\ z''(t) &= \frac {\partial F}{\partial z} \end {array} } \]
Where \(F=F(r)\) and \(r=\sqrt {x(t)^2+y(t)^2+z(t)^2}\)
Mathematica: cpu = 0.011001 (sec), leaf count = 137 \[ \left \{\left \{x(t)\to c_1 e^{-\frac {t \sqrt {f'(r)}}{\sqrt {r}}}+c_2 e^{\frac {t \sqrt {f'(r)}}{\sqrt {r}}},y(t)\to c_3 e^{-\frac {t \sqrt {f'(r)}}{\sqrt {r}}}+c_4 e^{\frac {t \sqrt {f'(r)}}{\sqrt {r}}},z(t)\to c_5 e^{-\frac {t \sqrt {f'(r)}}{\sqrt {r}}}+c_6 e^{\frac {t \sqrt {f'(r)}}{\sqrt {r}}}\right \}\right \} \]
Maple: cpu = 0.078 (sec), leaf count = 101 \[ \left \{ \left \{ x \left ( t \right ) ={\it \_C5}\,{{\rm e}^{{t\sqrt {{ \frac {\rm d}{{\rm d}r}}F \left ( r \right ) }{\frac {1}{\sqrt {r}}}}}}+ {\it \_C6}\,{{\rm e}^{-{t\sqrt {{\frac {\rm d}{{\rm d}r}}F \left ( r \right ) }{\frac {1}{\sqrt {r}}}}}},y \left ( t \right ) ={\it \_C3}\,{ {\rm e}^{{t\sqrt {{\frac {\rm d}{{\rm d}r}}F \left ( r \right ) }{\frac {1}{\sqrt {r}}}}}}+{\it \_C4}\,{{\rm e}^{-{t\sqrt {{\frac {\rm d}{ {\rm d}r}}F \left ( r \right ) }{\frac {1}{\sqrt {r}}}}}},z \left ( t \right ) ={\it \_C1}\,{{\rm e}^{{t\sqrt {{\frac {\rm d}{{\rm d}r}}F \left ( r \right ) }{\frac {1}{\sqrt {r}}}}}}+{\it \_C2}\,{{\rm e}^{-{t \sqrt {{\frac {\rm d}{{\rm d}r}}F \left ( r \right ) }{\frac {1}{\sqrt { r}}}}}} \right \} \right \} \]