I think your problem is the initial condition z(0) = 0 which make dz/dt(0) blow up for any nonzero Phi1 and Phi2(0)=0
Hello Kay. Thanks for your comment. No, the initial condition z(0)=0 is not a problem because initially the first derivative tends to infinity; that's compatible with the physical problem. The way I by-passed z(0)=0 is to consider z(0) ~ $10^{-15}$. Regards Javier
Hello Daniel, Thanks for your comments. Sometimes you have the answer in front of you and you don't realize. Yes the derivative works for some particular situations. Thanks, Regards Javier
P.S. : I upload the original Mathematica file
(1) Why not differentiate the second equation to get rid of the integral operator?
(2) Always better if the equations provided are in Mathematica input form. Then people can cut and paste them for purposes of attempting to answer questions.