\[ \left \{x''(t)=\frac {x(t) f'(r)}{r},y''(t)=\frac {y(t) f'(r)}{r},z''(t)=\frac {z(t) f'(r)}{r}\right \} \] ✓ Mathematica : cpu = 0.0075644 (sec), leaf count = 137
DSolve[{Derivative[2][x][t] == (x[t]*Derivative[1][f][r])/r, Derivative[2][y][t] == (y[t]*Derivative[1][f][r])/r, Derivative[2][z][t] == (z[t]*Derivative[1][f][r])/r},{x[t], y[t], z[t]},t]
\[\left \{\left \{x(t)\to c_1 e^{-\frac {t \sqrt {f'(r)}}{\sqrt {r}}}+c_2 e^{\frac {t \sqrt {f'(r)}}{\sqrt {r}}},y(t)\to c_3 e^{-\frac {t \sqrt {f'(r)}}{\sqrt {r}}}+c_4 e^{\frac {t \sqrt {f'(r)}}{\sqrt {r}}},z(t)\to c_5 e^{-\frac {t \sqrt {f'(r)}}{\sqrt {r}}}+c_6 e^{\frac {t \sqrt {f'(r)}}{\sqrt {r}}}\right \}\right \}\] ✓ Maple : cpu = 0.136 (sec), leaf count = 101
dsolve({diff(diff(x(t),t),t) = D(F)(r)/r*x(t), diff(diff(y(t),t),t) = D(F)(r)/r*y(t), diff(diff(z(t),t),t) = D(F)(r)/r*z(t)})
\[\left \{x \left (t \right ) = c_{5} {\mathrm e}^{\frac {\sqrt {\frac {d}{d r}F \left (r \right )}\, t}{\sqrt {r}}}+c_{6} {\mathrm e}^{-\frac {\sqrt {\frac {d}{d r}F \left (r \right )}\, t}{\sqrt {r}}}, y \left (t \right ) = c_{3} {\mathrm e}^{\frac {\sqrt {\frac {d}{d r}F \left (r \right )}\, t}{\sqrt {r}}}+c_{4} {\mathrm e}^{-\frac {\sqrt {\frac {d}{d r}F \left (r \right )}\, t}{\sqrt {r}}}, z \left (t \right ) = c_{1} {\mathrm e}^{\frac {\sqrt {\frac {d}{d r}F \left (r \right )}\, t}{\sqrt {r}}}+c_{2} {\mathrm e}^{-\frac {\sqrt {\frac {d}{d r}F \left (r \right )}\, t}{\sqrt {r}}}\right \}\]