Q:

Can solutions that we get from a direction field of a Linear ODE cross each other

Question:
When we sketch the direction fields of a given Linear ODE, can they cross each other ?
Reasoning:
Let we have a linear ODE of the form
$$y'(x) = f(x,y(x))$$
This means that for a given $(x,y)$, $f(x,y)$ have a definite value, hence at that point, the derivative of any function that satisfies the linear ODE is that unique value $f(x,y)$. Therefore, if at some point we have more than one solution, the solutions cannot cross each (they can meet, but not cross) other because that would violate the uniqness of the derivative at that point as I have argued.
Now, I have asked the same question to one of my professors by giving the above argument, but he said that "we don't know the form of $f(x,y)$, so we cannot sure such a thing", however, my argument directly eliminates his argument in my view, but I want to make sure that the above argument is a valid argument and my conclusion is correct.

A:

As I have explain in my question:

This means that for a given $(x,y)$, $f(x,y)$ have a definite value,
  hence at that point, the derivative of any function that satisfies the
  linear ODE is that unique value $f(x,y)$. Therefore, if at some point
  we have more than one solution, the solutions cannot cross each (they
  can meet, but not cross) other because that would violate the uniqness
  of the derivative at that point as I have argued.

