Solutions of a homogeneous differential equation

Question

Consider a homogeneous differential equation $\dfrac{dx}{dt} = f\left(\dfrac{x}{t}\right)$, where $f(k)=k$ for some $k \in \mathbb{R}$. Show that

1. If $f'(k) < 1$ then there is no solution tangent to $x(t) = kt$ at the origin except the obvious one $x(t)=kt$.
2. If $f'(k) > 1$ then there is an infinite number of such solutions.

The author gives a hint:

Two functions $y(t)$ and $x(t)$ defined for $t > 0$ are said to be tangent at $0$ if $\lim\limits_{t \to 0} \dfrac{y(t) - x(t)}{t} = 0$

I found some hints for 1. here, but the answer given is not clear to me and the author's hint is not used, so I'm still stuck on the problem. For 2. I have no idea.