Does a solution to the following integral exist?
$$\int \frac{y'}{x} dx. $$
I do not understand if have a solution without the integral. I'd like to write the solution in function of $y$, $x$, $y'$, $y''$, and so on. I put it on the Wolfram Alpha site and the Mathematica software and does not have any solution. So, I was trying something like that with Rules' chain $\dfrac{dy}{dx} = \dfrac{dy}{du} \dfrac{du}{dx}$ and consider $u = ln x$ implies $du = \frac{1}{x} dx$, we have that $\int \frac{y'}{x} dx = \int \frac{dy}{dx}\frac{1}{x} dx = \int \frac{dy}{dx}du = \int \frac{dy}{du}\frac{du}{dx}du$. But this do not help me.
Without having a specific form for $y$ there's no reason why this integral should result in anything specific or easily simplifiable. For example, if $y = e^{x}$ then this is essentially the Exponential integral $\textrm{Ei}(x)$ which is known to not be expressible in terms of elementary functions.
There are plenty of ways you could transform the integral into different forms. For example, you could apply integration by parts to get $\int \frac{y'}{x} dx = \frac{y}{x} + \int \frac{y}{x^2} dx$, but again it's going to depend a lot on what $y$ is as to whether this provides a simplification, or something that might provide some kind of convergent series, or whatever.