Suppose we have a second-order homogeneous ODE represented by $$ F(x,y,y',y'')=0 \quad (1), $$ and then consider the $4th$ order homogeneous ODE
\begin{equation} F(t,F(x,y,y,y''),F(x,y,y',y'')',F(x,y,y',y'')'')=0. \quad (2) \end{equation}
I want to know about the relation between the solutions of (1) and (2). For example, since they are homogeneous, I know that solutions of (1) are all solutions of $(2)$.
Particularly, I want to know in which cases there are other solutions of $(2)$ that are not solutions of $(1)$.