I'm solving the below differential equation:
3z^2 U'(z) -2 U(z)^2 - z U(z) + 2 = 0
I have two boundary conditions - $U(0)=1$, U'(0) = -\frac{1}{4}, however it is apparent that the set of solutions to this equation, before imposing the initial conditions, satisfy the first initial condition $U(0)=1$. Looking at the differential equation this is because it can be rearranged to:
$ [U(0)]^2 = 1 $
So it is evident that all of the general solutions should fit this condition. What's more, through "numerical experiment", I believe that the first derivative condition should be satisfied as well (though I can't prove it from the differential equation).
I am not sure that I can set the constant of integration to be anything I want, as this changes the behaviour of the solution. However, I can simplify the solution greatly by choosing "convenient" values of this constant.
Also the resulting function is a function of $\frac{1}{z}$, so it cannot actually be evaluated at 0, so I have been using limits, which feels quite mathematically dodgy.
This is a peculiarity that I have not come up against before - how exactly am I to interpret this?