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:
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]