When I had studied differential equations with separable variables at university in 1993, I had used category classification. Does anyone know any reference texts on this question? Here there is an explanation:
Let $X(x)$ and $Y(y)$ are two real functions defined in the intervals $(a,b)$ and $(c,d)$ respectively. The differential equation $$y'= X(x)\cdot Y(y) \tag 1$$ is defined as separate variables.
The first category solutions are those for which $Y(y(x))=0, \forall x\in(\alpha,\beta)$.
Second category solutions are those for which $Y(y(x))\neq 0, \forall x\in(\alpha,\beta)$.
The third category is a junction of the first and second category. Let be $\mathcal F_1$ the family of second category solutions defined in an interval $(\alpha,\beta[ \subset(\alpha,b)$ and such that
$$\lim_{x\to \beta^-}Y(y(x))=0 \tag 2$$
Let be $\mathcal F_2$ the family of solutions of the second category defined in an interval $]\alpha,\beta)\subset(a, \beta)$ and such that
$$\lim_{x\to \alpha^+}Y(y(x))=0 \tag 3$$
Let be
$$\mathcal F = \mathcal F_1 \cup \mathcal F_2$$ This is of course a third category of solutions.
Precisely solutions $y(x)$ such that $Y(y(x))$ takes in $(\alpha,\beta)$ both null and non null values. For the search of the solutions of the third category it is worth to observe that each solution of the third category is an extension of a function of $\mathcal F$. Let's say $y(x)$ a solution of the third category; it is precisely $Y(y(x_0))=0$ and $Y(y(x_1))\neq 0$ with $x_0 < x_1$ to fix the ideas. Let be $\mathcal Z$ the set formed by the numbers $x$ belonging to $[x_0,x_1[$ such that $Y(y(x))=0$. Let be $x_2 = \sup \mathcal Z$. The number $x_2 \in \mathcal Z$ (and therefore $x_2 = \max \mathcal Z$): the thing is obvious if $x_2 = x_0$ and we try immediately proceeding for absurd if $x_2 > x_0$ (it is enough to use the theorem of the permanence of the sign and the second property of the superior extreme). The restriction of $y(x)$ to $]x_2,x_1]$ belongs to $\mathcal F_2$ and therefore to $\mathcal F$.
Question: How can I find (or exist a specific method) the third category solutions in a simpler way i.e. to find $\mathcal F_1$ and $\mathcal F_2$ supposing to have a differential equation with separable variables?
Addendum: Is it possible to have a complete explanation, please?
