Suppose $f:\mathbb{R}\to\mathbb{R}$ is a continuous differentiable function such that $f(r)=r,$ for some $r.$ Then how to show that
If f'(r) < 1, then the problem x'=f(x/t) has no other solution tangent at zero to $\phi(t)=rt, t>0$.
Tangent here means
$\lim_{t\to 0^{+}}\frac{\psi(t)-\phi(t)}{t}=0$
I could only prove that $\psi(0^+)=0,$ and \psi'(0^+)=r. The problem was to use the fact that f'(r) < 1.