You solved my puzzle. The definitions differ. 

The analitical solution is correct. 

To answer why numerical solutions does not agree with analitical solution, the second derivative 

eval(diff(rhs(de), z), de) 

which is

-K^2*a20^2*(1-a3(z)^2/a20^2)*a3(z)-K^2*a10^2*(1-a3(z)^2/a10^2)*a3(z)

It is not zero at the point when de=0 unless a10=a20. Numerical solver used may have trouble going through the minimum for some reason.  Increase the Digits to 100, the solver stops at the minimum.

Thanks a lot.