Consider the partial differential equation $\quad$ $ u_{tt}+u_{xxxx} + \cos x \cos u = 0$.
Which of the following statements is correct?
Let $ L u = u_{tt}+u_{xxxx} + \cos x \cos u $. So, the PDE can be written as $ L u = 0 $ and therefore, it is homogeneous.
Let $ L u = u_{tt}+u_{xxxx} $. So, the PDE can be written as $ L u = - \cos x \cos u $ and therefore, it is not homogeneous.
I think the second one is the right one. Could you explain a little?
Also the partial differential equation is semilinear and therefore it is also quasi-linear. Is this correct?