In his book on ordinary differential equations, Arnold says that, in general, equations which contain unknown functions and their derivatives are not differential equations. For example, Arnold says,
$$ \frac{dx}{dt}=x(x(t)) $$
is not a differential equation.
How can one solve this non differential equation?
Playing around.
If $x'(t) = x(x(t))$, $x''(t) = x'(t)x'(x(t)) = x(x(t))x(x(x(t))) $. By induction, $x^{(k)}(t) $ involves terms with $x(x(...x(t)...)$ nested up to $k+1$ deep.
If $x(t) = \sum_{n=0}^{\infty} a_n t^n $, $x'(t) = \sum_{n=1}^{\infty} na_n t^{n-1} = \sum_{n=0}^{\infty} (n+1)a_{n+1} t^{n} $.
We need $x(0) = 0$ for the composition to be defined.
Therefore $x'(0) =x(0) = 0 $, $x''(0) = 0 $.
By induction on $k$, $x^{(k)}(0) =0 $ for all $k$.
Therefore $x(t)$ can not be a infinite power series.
If $x(t) =b t^a $, $x'(t) =abt^{a-1} $ and $x(x(t)) =b(bt^a)^a =b^{a+1}t^{a^2} $, so we need $ab=b^{a+1}$ (so $a = b^a$ or $b = a^{1/a})$ and $a^2=a-1$, so that
$a =\frac12(1\pm\sqrt{1-4}) =\frac12(1\pm i\sqrt{3}) $.
For this,
$\begin{array}\\ t^a &=t^{\frac12(1\pm i\sqrt{3})}\\ &=t^{1/2}e^{\pm \ln(t)\frac12( i\sqrt{3})}\\ &=t^{1/2}\left(\cos(\ln(t)\frac12( \sqrt{3}))\pm i\sin(\ln(t)\frac12( \sqrt{3}))\right)\\ \end{array} $
From $b = a^{1/a}$ we get $b$, but I'm not going to work that out.
That is all I can come up with for now.