Find all functions $y:\mathbb{R}\to\mathbb{R}$ such that $$y'(x)=y\Big(\frac{1}{x}\Big)$$ we can differentiate both sides to get a homogenous second order Euler equation $$y''(x)=-\frac{1}{x^2}y'\Big(\frac{1}{x}\Big)=-\frac{1}{x^2}y(x)\implies x^2y''+y=0$$ which is doable.
My question is, am I missing something? From a short glance at WA it doesn't seem that all of the solutions of the Euler equation uphold the required criteria.