On the order of growth of transcendental entire solutions of algebraic differential equations of second order

The basic result is the following.
Theorem. A differential equation of the form $P(z,y,y',y'')=0$, where $P(z,u,v,w)$ is a polynomial in the variables $z$, $u$, $v$ and $w$, cannot be satisfied by entire transcendental functions of zero order of growth.
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