What are the two ODEs?

A complete explanation of the problem can be found bellow on the attached pdf. Thank you for your interest.

Thanks to Benoit Bardet, Nicholas Clauvelin and Masoomeh Taghipoor

Dear Yannick,

Please refer my articles. I have solved the system of odes.

Thank You Narendrakumar Dasre

Hi Yannick. You can use "shotting methods". It's explained in the book Applied Numerical Methods with MATLAB for Engineers - Steven Chapra, Chapter 24 (Boundary Value Problems). Another form of solution is to use finite difference methods, explained too in the same book Chapter 24.

Thank you Eberto Porto

Hi Yannick, if you want to use fast and reliable routine to implement in Matlab a shooting algorithm you can resort to the sundialsTB interface to the SUNDIALS suite (https://computation.llnl.gov/casc/sundials/main.html).

Thank you Federico Bizarri. That's great !!!

I believe nonlinear shooting method should work. T.Y.Na has explained the procedure in his book.

Thank you Mazhar Iqbal .

For readers that are mathematic fans, here is an example of solving a rather stiff BVP involving 3 coupled ODEs (for the velocity field using a shooting method :

In[92]:= ODE[z1_, z2_] := {2 F[x] + H'[x] == 0,

F[x]^2 + H[x] F'[x] - G[x]^2 == F''[x],

F[x] G[x] + H[x] G'[x] + F[x] G[x] == G''[x], G[0] == 1, F[0] == 0,

H[0] == 0, G'[0] == z2, F'[0] == z1};

sol[z1_, z2_] :=

NDSolve[ODE[z1, z2], {F[x], G[x], H[x]}, {x, 0, 10}];

FendBC[z1_?NumericQ, z2_?NumericQ] :=

First[(F[x] /. sol[z1, z2]) /. x -> 10];

GendBC[z1_?NumericQ, z2_?NumericQ] :=

First[(G[x] /. sol[z1, z2]) /. x -> 10];

bc = FindRoot[{FendBC[z1, z2] == 0, GendBC[z1, z2] == 0}, {z1, 0.49,

0.53}, {z2, -0.58, -0.62}, MaxIterations -> 500]

Out[96]= {z1 -> 0.510214, z2 -> -0.61591}

Here is a plot of the velocity field

I think that these problems are (mathematical) equivalent to Boundary Value Problems (BVPs). So you can use any method for solving BVP like FDM or FEM.