Not being an experienced person in differential equations, I would like to clarify some issues for the following type of equations.
Let we have the following equation
$$y' = \frac{g(y)}{f(x)},$$
where $g\in C^1[\alpha,\beta], f\in C^1[a,b]$ and both funtions are non-negative and $y(a)=\alpha$.
Rewriting the equation$$\frac{dy}{g(y)} = \frac{dx}{f(x)},$$ it is easy to see that the general integral of the equation is
$$\int^y_{y_0}\frac{dy}{g(y)} - \int^x_{x_0}\frac{dx}{f(x)}= const.$$
If I take $\alpha=y_0=y(x_0)=y(a)$ then I'll have$$\int^y_{\alpha}\frac{dy}{g(y)} = \int^x_{a}\frac{dx}{f(x)}.$$
If I add a condition $y(b)=\beta$ to the problem above then in fact I want to find a solution which is a diffeomorphism between $[a,b]$ and $[\alpha,\beta]$ (since $y' = \displaystyle\frac{g(y(x))}{f(x)}>0$ for all $x$).
Potentially, if we assume that the original equation is globally solvable, then the criterion for a solution to be a diffeomorphism could be the equality of the above integrals, BUT what other conditions are needed to guarantee existence?
Implicit function theorem gives only local solution from the general integral.
Picard's theorem requires strict restrictions on the Lipschitz constant and the bounds of the segments.
Are there other ways to guarantee the existence and uniqueness of a solution to such equations?