If the zero function satisfies a PDE, then is the PDE homogeneous?

Question

In my lecture notes, it clearly states that for a general PDE $F(x,t,u,u_x,u_t,...)=0$, if $u(x,t)=0$ satisfies the PDE then the PDE is homogeneous. However, it seems that the definition of a homogeneous PDE is that if $u(x,t)$ is a solution then $\alpha u(x,t)$ is also a solution, for any $\alpha$. I'm seeking clarity on this? Since the result given in my lecture notes does not satisfy the definition (if this is the correct definition?), for example with the PDE $u_{tt}+uu_{xx}=0$?

Answer 1

I suspect the author was thinking of homogeneous linear equations.