The solution and plot of the Differential equation is $\frac{d^2S(r)}{dr^2}+\frac{1}{r}\frac{dS(r)}{dr}-S(r)+3S^3(r)=0$

S0 = S[r] /.
NDSolve[{S''[r] + 1/r S'[r] - S[r] + S[r]^3 == 0,
S[1/10000000000000000] == 2.20620086,
S'[1/10000000000000000] == 0}, S, {r, 1/1000, 15}];
Plot[{S0}, {r, 1/100, 14}, PlotRange -> Full]


I have to use the above info in the differential equation:

$\frac{d^2Z(r)}{dr^2}+\frac{1}{r}\frac{dZ(r)}{dr}-Z(r)+3S^2Z(r)-S^5(r)=0$

And I have plotted like this: but it is not working

sol = NDSolve[{s''[r] + 1/r s'[r] - s[r] + s[r]^3 == 0,
s[1/10000000000000000] == 2.20620086,
s'[1/10000000000000000] == 0}, s, {r, 1/1000, 15}];
sol = s[r] /. sol;
Z0 = Z[r] /.
NDSolve[{Z''[r] + 1/r*Z'[r] - Z[r] + 3 sol^2*Z[r] - sol^5 == 0,
Z[1/10000000000000000] == -16.17403,
Z'[1/10000000000000000] == 0}, Z, {r, 1/1000, 15}];
Plot[{Z0}, {r, 1/100, 14}, PlotRange -> Full]


The output curve will be like: .

I have used r instead of $\rho$. I have tried several times to get the output but it shows some following which I don't understand. I have looked the error to get but couldn't.

Sorry for the poor question. I have studied but couldn't fix it.

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Hats off to you for this persistent tsunami of gibberish questions! When it comes to learn from others answer and comments about your post you show no positive effort whatsoever...Waiting for the light of resurrection here... –  PlatoManiac Sep 9 '13 at 12:50

Your coupled differential equations has nothing to do with linking. The appropriate word should be "coupled" not "linking", which can be solved very easily some thing like this

sol = NDSolve[{s''[r] + 1/r s'[r] - s[r] + 3 s[r]^3 == 0,Z''[r] + 1/r Z'[r] - Z[r] + 3
s[r]^2 Z[r] - s[r]^5 == 0, s[1/10000000000000000] == 2.20620086, s'[1/10000000000000000] == 0
,Z[1/10000000000000000] == -16.17403, Z'[1/10000000000000000] == 0}, {s, Z}, {r, 1/1000, 15}]


Now for plotting we can use

Plot[{s[r] /. sol, Z[r] /. sol}, {r, 1/1000, 15}, PlotRange -> All,

PlotStyle -> {Blue, Red}, PlotLegends -> Placed[{"s[r]", "Z[r]"}, Above]]


The output is

which clearly satisfy your condition on s and z.

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