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In Communications in Contemporary Mathematics 24 3, (2022),the authors have developed a method for constructing G-invariant partial differential equations (PDEs) imposed on hypersurfaces of an 
$(n+1)$-dimensional homogeneous space 
$G/H$, under mild assumptions on the Lie group G. In the present paper, the method is applied to the case when 
$G=\mathsf{PGL}(n+1)$ (respectively, 
$G=\mathsf{Aff}(n+1)$) and the homogeneous space 
$G/H$ is the 
$(n+1)$-dimensional projective 
$\mathbb{P}^{n+1}$ (respectively, affine 
$\mathbb{A}^{n+1}$) space, respectively. The main result of the paper is that projectively or affinely invariant PDEs with n independent and one unknown variables are in one-to-one correspondence with invariant hypersurfaces of the space of trace-free cubic forms in n variables with respect to the group 
$\mathsf{CO}(d,n-d)$ of conformal transformations of 
$\mathbb{R}^{d,n-d}$.
