The appearance of the Riccati equation when we applied scale relativity principle by L.Nottale on some quantum systems , this lead to believe that Scale relativity theory ScR is sweetable for quantum physics.
In mathematics, a Riccati equation is any ordinary differential equation that has the form R'=q0(x) +q1(x)R+q2(x)R^2 where q0,q1 and q2 are known functions.
to let this question more clear, when we solved any quantum problems like free particle, finite well , simple harmonic and ...ect, mathematically then we will get an equation have Riccati form then change it to 2nd differential equation to complete our solution. So our above asking why Reccati equation appear always in quantum problems, I think there is a deep connection between them.