Consider the equation
$$ y' = a_0(x) + a_1(x)y + a_2(x)y^2 + a_3(x)y^3 $$
Does there exist a substition $y = f(a_0, a_1, a_2, a_3, u, u', u'')$ such that after simplifying from this subsitution we get an equation of the form
$$ u''' = b_0(x) + b_1(x)u + b_2(x)u' + b_3(x)u'' $$
My Motivation for why such an substitution should exist even if non-elementary
The solution to a first order differential equation takes on the form $y = F(x,C)$ for some function F and a constant factor. This particular differential also happens to be a third degree polynomial meaning that there exist 3 unique first order equations that can be generated from it if we solve y as a function of y' and the coefficients.
nth order linear differential equations can vary by up to n constants in their solution and therefore a 3rd order linear differential equation should have enough degrees of freedom to model any third-degree polynomial differential equation such as the one I have given.
This is evidenced by the fact that second degree polynomial differential equations (THAT IS RICCATI EQUATIONS) are known to be convertible to second degree linear differential equations.