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?