I have a set of differential equations:
w1'[u] == -1 - 1/w1[u] + 1/w2[u] + 1/w3[u],
w2'[u] == -1 - 1/w1[u] +1/w2[u] + 1/w3[u],
w3'[u] == -1 - 1/w1[u] + 1/w2[u] + 1/w3[u],
w2R'[u] == -1 - 1/w2R[u] + 1/w2L[u] + 1/w4C[u],
w2L'[u] == -1 - 1/w2R[u] +1/w2L[u] + 1/w4C[u],
w4C'[u] == -1 - 1/w2R[u] + 1/w2L[u] + 1/w4C[u],
and in addition to some some normal boundary conditions
w1[Log[91.1876]] == 1/0.016887, w2[Log[91.1876]] == 1/0.03322,
w3[Log[91.1876]] == 1/0.12
the boundary conditions for w2R,w2L,w4C are a bit more variable. Firstly, they should meet at a point $M$, which should be computed and is a priori unknown:
w2R[M]==w2L[M]==w4C[M]
and secondly at some other, a priori unknown point M2 the solutions should fulfil
w1[M2]== w2R[M2] +w4C[M2]
w2[M2]== w2R[M2]
w3[M2]== w4C[M2]+3
How I can get the solutions of this system, i.e. M,M2, w1, w2,w3,w2R,w2L,w4C?
The system for w1,w2,w3 can be solved individually with NSolve, but my problem is how to solve the system for w2R,w2L,w4C, with the boundary conditions dependent on M and M2 that should be computed by Mathematica, too.
For a simpler set of differential equations I could simply solve, for example
w1'[u] == -1 ,
w2'[u] == -1 ,
w3'[u] == -1,
w1[Log[91.1876]] == 1/0.016887, w2[Log[91.1876]] == 1/0.03322,
w3[Log[91.1876]] == 1/0.12
analytically using DSolve and likewise
w2R'[u] == -1,
w2L'[u] == -1 ,
w4C'[u] == -1 ,
with DSolve without boundary conditions. Then the analytic solutions for w2R,w2L and w4C depend on three constants C1, C2, C3. Then I can compute the unknowns C1, C2, C3, MI and MU using ordinary NSolve and the boundary conditions
w2R[M]==w2L[M]
w2R[M]==w4C[M]
w1[M2]== w2R[M2] +w4C[M2]
w2[M2]== w2R[M2]
w3[M2]== w4C[M2]+3
Unfortunately, this approach fails for more complicated differential equations that can't be solved analytically.
w2R[M]==w2L[M]==w4C[M]
is equivalent tow2R[M]==w2L[M]
andw2L[M]==w4C[M]
, while you have three dependent variables. $\endgroup$