BVP system of nonlinear coupled ODEs

Here I am, trying to solve this system of coupled ODEs (up to a minus sign):

$u''=6u^5-(8+4a)u^3+(2+4a)u+\frac{2u((w^2-s)^2+bw)}{(u^2+c)^2}$

$w''=\frac{4w^3-4bw+b}{u^2+c}$

with the boundary values being $u(0)=0, u(7)=1, w(0)=s, w(7)=-s$

and the parameters being $a=1/10, b=1/10, c=1/20, s=1.1$.

Now I want to use the shooting method to solve these problems, but all I did in the interim was to write a code to solve the problem via conventional NDSolveValue:

eps = 1/10;
eps2 = 1/10;
eps3 = 1/20;
anti = 1.1;
{phi6m, phi6n} =
NDSolveValue[{D[u[x], x, x] == -(-6 u[x]^5 + (8 + 4 eps) u[x]^3 - (2 + 4 eps) u[x] +
(v[x]^4 - 2 *anti*v[x]^2 + eps2*v[x] + anti^2)*2 u[x]/(u[x]^2 + eps3)^2),
D[v[x], x, x] == (4 v[x]^3 - 4 *anti*v[x] + eps2)/(u[x]^2 + eps3),
u == 0, Derivative[u] == 0.3, v == anti,
Derivative[v] == -0.005}, {u, v}, {x, 0, 7}]
Plot[phi6m, {x, 0, 0.3}, PlotRange -> All]
Plot[phi6n, {x, 0, 0.3}, PlotRange -> All]
• Then you'll have a hard time looking for a proper "StartingInitialConditions"……there're many examples for shooting method in this site, you can have a search. – xzczd Apr 1 '14 at 3:19