Does any iterative equation of n-th order which does not inclute derivatives of order higher than 1 have exactly n independent solutions?

Let's designate n-th iterate of a function $y(x)$ as $y^{[n]}(x)$

Is it true that the equation

$$F(y^{[n]},y^{[n-1]},...,y,x,y')=0$$

has exactly n independent solutions?

A method of solving some classes of iterative equations was proposed in this paper: http://faculty.kfupm.edu.sa/math/akca/papers/cheng.pdf

It gives exactly n indepentent solutions for each solved iterative equation of n-th order.

For example, the equation

$$y^{[2]}-y'=0$$

has exactly two solutions:

$y_1(x) = e^{\frac{\pi}{3} (-1)^{1/6}} x^{\frac{1}{2}+\frac{i \sqrt{3}}{2}}$

$y_2(x) = e^{\frac{\pi}{3} (-1)^{11/6}} x^{\frac{1}{2}+\frac{i \sqrt{3}}{2}}$

The question here is it the property of any iterative equation, not just of those which suitable for this method.