\begin{align}

&a (x,t) u_t + b(x,t) u_x + c(x,t)u=0 && \text{linear homogeneous},\\[3pt]

&a (x,t) u_t + b(x,t) u_x + c(x,t)u=f(x,t) && \text{linear inhomogeneous},\\[3pt]

&a (x,t) u_t + b(x,t) u_x + c(x,t)u=f(x,t,u) && \text{seminilear},\\[3pt]

&a (x,t,u) u_t + b(x,t,u) u_x = f(x,t,u) && \text{quasilinear},\\[3pt]

&F(x,t,u,u_t,u_x)=0 && \text{nonlinear}

\end{align}

and each type is more general than the previous one.

Writing corresponding ODE we get respectively

\begin{align}

&\frac{dx}{a}=\frac{dy}{b}=-\frac{du}{cu} \implies &&\phi(x,t)=C_1, u=C_2\psi(x,t)\tag{1},\\

&\frac{dx}{a}=\frac{dy}{b}=-\frac{du}{cu} \implies &&\phi(x,t)=C_1, u=C_2\psi(x,t)+g(x,t)\tag{2},\\

&\frac{dx}{a}=\frac{dy}{b}=-\frac{du}{c} \implies &&\phi(x,t)=C_1, \psi(x,t,u)=C_2\tag{3},\\

&\frac{dx}{a}=\frac{dy}{b}=-\frac{du}{c} \implies &&\phi(x,t,u)=C_1, \psi(x,t,u)=C_2\tag{4},\\

\end{align}

For nonlinear see

http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter2/S2.2.html#sect-2.2.2, not covered in the course