What is the meaning of this differential equation-ever thought about it?

Question

Consider a differential equation $y'^2+xy\operatorname{ln}(xy'')=\sin(1/xy)$,what on earth does it actually mean.I mean consider $dy/dx=y$,it simply means that we have to find $y=f(x)$ defined on $\mathbb R$ such that $d/dx(f(x))=f(x)$.But what does the first differential equation mean.There we have an implicit and complicated relation between $x,y,y',y''$ etc and often the solutions to the differential equations are not an explicit function of $x$ such as $y=f(x)$.But we get some implicit solution $\phi(x,y)=0$.So what does it mean by the differential equation I mentioned in the beginning.It probably may mean that find $y=f(x)$ such that the expression holds.And its solution means all functions $y=f(x)$ such that $\phi(x,f(x))=0$.Am I right? Usually these differential equation courses are taught pretty mechanically without analyzing things as we do in case of real analysis or other pure maths topics.I want to study this differential equation with an analytic approach.Can someone help me. I would also accept any good reference books that might be helpful.