This is a question on global existence and uniqueness for an IVP. Consider the following first-order nonlinear ODE (known as a Ricatti equation):
\begin{align} y^{\prime} = f_1(x) y^2 + f_2(x) y + f_3(x),\qquad x\in [0,\infty)\tag{1}, \end{align}
subject to a boundary condition
\begin{align} y(0) = y_0.\tag{2} \end{align}
Let $f(x,y) = f_1(x) y^2 + f_2(x) y + f_3(x),$ i.e., let $f:[0,\infty)\times (-\infty,\infty)\rightarrow (-\infty, \infty)$ be the right hand side of (1). Assume that $f$ is locally Lipschitz in $y$. From this post, it appears that local Lipschitzness guarantees that a global solution is unique iff a global solution exists.
My question is the following: Do Ricatti equations (in particular those defined for an unbounded time domain) always have solutions? Presumably, if my statements were correct, then (1)-(2) would have a unique global solution. I ask this question because I have a problem with only local Lipschitzness, not global, so existence is an issue. Some IVPs have solutions that blow up in finite time and cannot be extended globally. I am hoping that in the case of this particular type of ODE this is not an issue.