Q:

Solving a System of Differential Equations for Pendant Drop Application

So I'm trying to solve the system of differential equations describing a pendant drop. The system is as follows:
system = {
 ϕ'[s] == 2 + β*z[s] - Sin[ϕ[s]]/x[s], 
 x'[s] == Cos[ϕ[s]],
 z'[s] == Sin[ϕ[s]],
 x[0] == 0, 
 z[0] == 0,
 ϕ[0] == 0
};

Beta is a constant for the liquid. There are a few other versions of these equations out there. Anyways the literature I've found sound far has solved these numerically and displays these solutions in tables or just says that the solution is obtainable. I was hoping I might be able to find a way to do that more efficiently here. 

A:

Try  ParametricNDSolve with slightly modified initial value x[0]==.0001 (see comment @user21 ) 
lsg = ParametricNDSolveValue[{\[Phi]'[s] == 2 + \[Beta]*z[s] - Sin[\[Phi][s]]/x[s], x'[s] == Cos[\[Phi][s]], z'[s] == Sin[\[Phi][s]], x[0] == 0.001, z[0] == 0, \[Phi][0] == 0}, {x, z, \[Phi]}, {s, 0, 1}, {\[Beta]}]    

For example lsg[.1][[1]][s] gives the solution x[s] with beta=.1.
Plot[{lsg[#][[1]][s], lsg[#][[2]][s], lsg[#][[3]][s]}, {s, 0, 1}] &[.1]     

