Certainly the problem can be solved numerically (FD or FE), and I believe analytically, but I'd have to dig back in my archives for that.
one could write \nu_1 and \nu_2 in thex LaTeX expressions before \Sigma.
I think this is how the equations are supposed to look:
[tex]{D_1}\frac{{d^2}\phi_1}{dx^2}\,+\,\Sigma_{R1}\phi_1\,=\,\frac{1}{k}({\n u_1}{\Sigma_{f1}\phi_1}\,+\,{\nu_2}{\Sigma_{f2}\phi_2})[/tex]
[tex]{D_2}\frac{{d^2}\phi_2}{dx^2}\,+\,\Sigma_{a2}\phi_2\,=\,{\Sigma_{s12}}\ phi_1[/tex]
Just looking these, one could collect coefficents and rewrite the equations as:
[itex]\phi_1[/itex]'' + A [itex]\phi_1[/itex] = B [itex]\phi_2[/itex]
[itex]\phi_2[/itex]'' + C [itex]\phi_2[/itex] = D [itex]\phi_1[/itex]
