This is an old question but I am having a new problem. I was able to solve z,r,phi as a function of s but what I really need at the end is to plot z as a function of r. Since dz/dr=Tan[phi], I need to integrate Tan[phi] with respect to r. How should I do this?

With[{Bo = 5},
Eqns = {2 - Bo z[s] - Sin[φ[s]]/r[s] == 
Derivative[1][z][s] == Sin[φ[s]], 
Derivative[1][r][s] == Cos[φ[s]],
z[0] == 0, φ[0] == 0, r[.001] == .001}];

sol = NDSolve[Eqns, {z, r, φ}, {s, 0.01, 3}]
f1 = (Tan[φ[s]] /. sol);
f11 = NIntegrate[f1, {r, 0, 3}];
p1 = Plot[f11, {r, 0.01, 3}, PlotStyle -> Red]



You can use ParametricPlot:

sol = NDSolveValue[Eqns, {z, r, \[CurlyPhi]}, {s, 0.01, 3}] 
ParametricPlot[ {sol[[2]][s], sol[[1]][s]} , {s, 0.01, 3},AxesLabel -> {r[s], z[s]}]

enter image description here

  • $\begingroup$ For some reason this does not work for me. The plot has the axes but no graph. $\endgroup$
    – Alex Wu
    Feb 9 '18 at 17:32
  • $\begingroup$ I edited my answer, hope it works now! $\endgroup$ Feb 10 '18 at 10:52

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