I am attempting to solve the Ricatti equation:
$$ 3x^2 y' - 2y^2 - x y +2=0$$
And I have attempted the method suggested on the Wikipedia page of converting it to a linear second order differential equation. I have solved this second order ODE, but this comes with two solutions. That is, calling our solution $u(x)$ I have solved it to find
$$ u(x) = \alpha u_1(x) + \beta u_2(x) $$
However, when re-substituting to find the equation y(x), I will still have 2 unknown quantities (both $\alpha$ and $\beta$), with only one initial condition. It strikes me that there must be some relationship between $\alpha$ and $\beta$, as the original equation is only first order and should only require one initial conditions. However, I cannot find any literature on this.
Is my supposition correct? And if so, what is the relationship between the two constants?