# 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.

• Hi Grace, welcome to Mma.SE. Start by taking the tour now and learning about asking and what's on-topic. Always edit if improvable, show due diligence, give brief context, include minimal working example of your code and data in formatted form. By doing all this you help us to help you and likely you will inspire great answers. The site depends on participation, as you receive give back: vote and answer questions, keep the site useful, be kind, correct mistakes and share what you have learned. – rhermans Jul 10 '19 at 15:11
• Try NDSolve but watch out that at s==0 you divide by 0 with /x[s]. – user21 Jul 10 '19 at 15:13

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]