The definition of semilinear equation in 2.1 of textbook

[Luyu CEN]

In the textbook, it reads

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.

In this case the last ODE is also linear.

But I don't understand this sentence. Is there an ODE?

[Victor Ivrii]

In the textbook, it reads
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.

It may be linear, but generally semilinear:
$$a(x,t)u_t+b(x,t)u_x=f(x,t,u)$$

Definition 3. Furthermore if f is a linear function of u: f=c(x,t)u+g(x,t) original equation is linear.[/center]
In this case the last ODE is also linear.
But I don't understand this sentence. Is there an ODE?

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.