Is this PDE $u_t+u_{xxxx}+\sqrt{1+u}=0$ nonlinear homogeneous?

Question

Consider the PDE \begin{gather*}\tag{1} u_t+u_{xxxx}+\sqrt{1+u}=0, \end{gather*} where $u=u(x,t),$ which occurs in Page 5 of Strauss's book of PDE. I am wondering if it is homogeneous. Nonlinearity of this PDE is apparent. How can I determine if a given PDE is homogeneous, especially it is nonlinear? What is the formal definition of homogeneous PDE, no matter what it is linear, or nonlinear? How about this PDE \begin{gather*}\tag{2} u_t+u_{xxxx}+\sqrt{x^2+t^2+u}=0, \end{gather*} whether it is homogeneous, or inhomogeneous?

Answer 1

The point of homogeneity, if you're not considering linear problems necessarily, is not simply "you can put everything involving the solution on one side and the other side is zero". The point of homogeneity is that any multiple of a solution is a solution. This does not hold for either of those equations.