In the textbook, it readsIt may be linear, but generally semilinear:
Definition 2. If a=a(x,t) and b=b(x,t) equation is semilinear.
But in the previous section this should be linear equation with variable cofficients, right?
I think the next definition is consistent with what I know.
Definition 3. Furthermore if f is a linear function of u: f=c(x,t)u+g(x,t) original equation is linear.[/center]After we found integral curves $x=x(t,c)$ satisfying $\frac{dt}{a(x,t)}=\frac{dx}{b(x,t)}$ we have ODE $\frac{du}{dt}=\frac{df(x,t,u)}{a(x,t)}$ with $x=x(t,c)$ along integral lines. For semilinear PDE this ODE is non-linear, for linear PDE this ODE is linear.
In this case the last ODE is also linear.
But I don't understand this sentence. Is there an ODE?