My textbook has classified the following PDEs accordingly:
$$u_{xxx} - 4u_{xxyy} + u_{yyzz} = f(x, y, z) \ \ \ \text{Linear}$$
$$u^2_{x}u_{tt} - \dfrac{1}{2}u = 1 - u^2 \ \ \ \text{Semi-Linear}$$
$$u_{tt}u_{xxx} - u_{x}u_{ttt} = f(u, x, t) \ \ \ \text{Quasi-Linear}$$
$$\exp(u_{xtt}) - u_{xt}u_{xxx} + u^2 = 0 \ \ \ \text{Fully Non-Linear}$$
$$2\cos(xt)u_t - xe^tu_x - 9u = e^t \sin(x) \ \ \ \text{Linear}$$
$$x\cos(t)u_t + t u_x + u^2 u = \dfrac{x}{t} u \sin(u) \ \ \ \text{Semi-Linear}$$
$$uu_t + u^2 u_x + u = e^x \ \ \ \text{Quasi-Linear}$$
$$u_t + \dfrac{1}{2} u_x^2 - u = \cos(xt) \ \ \ \text{Fully Non-Linear}$$
The textbook gives the following explanations of linear, semi-linear, quasi-linear, and fully non-linear PDEs:
Linear: In effect, all coefficients of $u$ and any of its derivatives must depend only on the independent variables.
Semi-linear: The coefficients of the highest derivatives of $u$ do not depend on $u$ or any derivatives of $u$.
Quasi-linear: The coefficients of the highest derivatives of $u$ depend only on lower derivatives of $u$.
Otherwise the PDE is fully nonlinear.
Reading through the classification of the aforementioned PDEs, I have a suspicion that there are some errors. I would greatly appreciate it if people could please review the author's classification of these PDEs and comment on its correctness.
I agree with you that this classification is not easy to follow. For example for $$u^2_{x}u_{tt} - \dfrac{1}{2}u = 1 - u^2 \ \ \ \text{Semi-Linear}$$ What is the highest derivative and what is its coefficient? If you say it is u_{tt}, then the coefficient depends on derivative of $u$ so it is not semi-linear.
I like to see the definition of highest derivative before getting into this classification.