You can solve any order ODE, PDE that is reasonably solvable in Mathematica: Notice I specify a function as r[t] and theta[t] and so on. Remember to cut/paste it then change it to Standard Form via Cell/Convert To/Standard Form so it's easier to read. This is however Mathematica 7. If you have an earlier version there may be downward incompatibilities.

Code:

k = 1;
m = 1;
g = 1;
eqn1 = Derivative[2][r][t] ==
(k/m)*(1 - r[t]) +
r[t]*Derivative[1][\[Theta]][t]^2*
Sin[\[Theta][t]] + g*Cos[\[Theta][t]];
eqn2 = Derivative[2][\[Phi]][t] ==
-2*Derivative[1][\[Phi]][t]*
(Derivative[1][r][t]/r[t] -
Derivative[1][\[Theta]][t]*Cot[\[Theta][t]]);
eqn3 = Derivative[2][\[Theta]][t] ==
Derivative[1][\[Phi]][t]^2*Sin[\[Theta][t]]*
Cos[\[Theta][t]] -
(2*Derivative[1][r][t]*
Derivative[1][\[Theta]][t])/r[t] -
(g/r[t])*Sin[\[Theta][t]];
mysol = NDSolve[{eqn1, eqn2, eqn3,
r[0] == 1, Derivative[1][r][0] ==
-2^(-1), \[Phi][0] == -Pi/2,
Derivative[1][\[Phi]][0] == 0,
\[Theta][0] == Pi/2, Derivative[1][\[Theta]][
0] == Pi}, {r, \[Phi], \[Theta]}, {t, 0, 1}]
Plot[Evaluate[{r[t], \[Phi][t], \[Theta][t]} /.
mysol], {t, 0, 1}, PlotStyle ->
{Red, Blue, Green}]