In Matlab, enter the following command:

doc bvp4c

This might be what you need.

Explicitly evaluate the partial derivatives of F w.r.t. a and b in the right hand side of your ODE. Write this expression as matlab function in dependence of the variable a.

Then applying any of Matlab's ODE solvers, e.g. ode23, ..... should do the trick.

Chebfun is a nice tool in Matlab with a GUI - you should give it a try...

http://www2.maths.ox.ac.uk/chebfun/

http://www.tem.uoc.gr/~danton/odeex.pdf

You can see some basic codes in my notes with their output, the notes are in greek but you can copy-paste the codes.

I send my answer another way as a file attachment too . Please, answer me. Your sincerely, Anna Tomova.

About the question if the solution of ODE a(k), a(0) = 1.8 has zero or not, my answer is: a(k) has'not 0 for every k, because F(0, b(0,k) = 0, but we have proved that if a(0) = 1.8, F (a, b(a, k ) = const = - 116. 051...

use ODE solver in MATLAB either ode45 or other one is most probable ode112