I am wondering what the essential value of an implicit solution to an (exact) ODE is? In ODE books usually implicit solutions occur in the context of exact differential equations, e.g.
$e^x\sin(x)+e^y\cos(y)\frac{dy}{dx}=0$ is an exact equation and the solution is given implicitly through the equation $\frac{1}{2}(\sin(x)-\cos(x))e^x+\frac{1}{2}(\sin(y)+\cos(y))e^y.$
In this example the solution can not be made explicit as it is often the case. Hence books usually leave it at that. But what do I gain with such an implicit solution? Sometimes it takes even a lot of effort to find an appropriate integrating factor first and then a potential function.
Therefore I suppose there must be at least some qualitative properties of the implicitly given solution which can be obtained from the algebraic equation in contrast to the defining ODE?!
Maybe someone can recommend good lterature where such questions are discussed?
I would welcome any help! Best wishes