\[ \left \{x''(t)=\frac {x(t) f'(r)}{r},y''(t)=\frac {y(t) f'(r)}{r},z''(t)=\frac {z(t) f'(r)}{r}\right \} \] ✓ Mathematica : cpu = 0.0067014 (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 = 0.194 (sec), leaf count = 101
\[ \left \{ \left \{ x \left ( t \right ) ={\it \_C5}\,{{\rm e}^{{t\sqrt {{\frac {\rm d}{{\rm d}r}}F \left ( r \right ) }{\frac {1}{\sqrt {r}}}}}}+{\it \_C6}\,{{\rm e}^{-{t\sqrt {{\frac {\rm d}{{\rm d}r}}F \left ( r \right ) }{\frac {1}{\sqrt {r}}}}}},y \left ( t \right ) ={\it \_C3}\,{{\rm e}^{{t\sqrt {{\frac {\rm d}{{\rm d}r}}F \left ( r \right ) }{\frac {1}{\sqrt {r}}}}}}+{\it \_C4}\,{{\rm e}^{-{t\sqrt {{\frac {\rm d}{{\rm d}r}}F \left ( r \right ) }{\frac {1}{\sqrt {r}}}}}},z \left ( t \right ) ={\it \_C1}\,{{\rm e}^{{t\sqrt {{\frac {\rm d}{{\rm d}r}}F \left ( r \right ) }{\frac {1}{\sqrt {r}}}}}}+{\it \_C2}\,{{\rm e}^{-{t\sqrt {{\frac {\rm d}{{\rm d}r}}F \left ( r \right ) }{\frac {1}{\sqrt {r}}}}}} \right \} \right \} \]