Consider the equation $u_y=u_x^3$. Show that the only solutions that are differentiable for all $(x,y)\in \mathbb{R}^2$ are linear functions $u(x,y)=ax+a^3y+b$ for some constants $a,b$.
By taking partial derivatives, I can show that the determinant of Hessian matrix is identically zero, but it is still not sufficient to show that the function is linear. I don't know how to proceed.