This regards Apostol Calculus 1 exercise 14 of section 13.25 .
In the exercise it is required to use the Cartesian equation of all conics of eccentricity $e$ and center at the origin to prove that these conics are integral curves of the differential equation $$y' = (e^2-1)\frac{x}{y}$$ This is easy enough because differentiating $\displaystyle\frac{x^2}{a^2}+\frac{y^2}{a^2(1-e^2)}=1$ gives $x(1-e^2)+yy' = 0$ which implies the given d.e.
But then there is a note which says :
Since this is a homogeneous differential equation ($y' = f(x,y) = f(tx,ty)$) the set of all such conics of eccentricity $e$ is invariant under a similarity transformation
I don't get why this is true, I understand that it is a homogeneous d.e. but I don't get the relation with invariance under a similarity transformation (replacing $x$ with $tx$ and $y$ with $ty$ in the equation of the ellipse).
Can someone clarify please? Thanks in advance