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 $f'(r) < 1.$