PDE Linear Equation Q: Homogeneous vs Nonhomogeneous

[Miike012]

My questions concerns the information in the document.
I highlighted the portion that is confusing me and a sample problem at the bottom.

Question:
Look at the equation 2.2.4 in the document.
When I set the function u equal to zero the equation becomes
0 = 0 + 0 + f(x,t) or f(x,t) = 0.

Now if you look in the document once more at the underlined green section is says that if f is equal to zero then the equation is homogeneous. However if you read the underlined orange section it says equation 2.2.4 is a nonhomogeneous equation...

This seems a bit contradictory. Some one please help me understand what they are saying.

 

[LCKurtz]

You have shown that L(0) = f(x,t). Since that isn't zero, the equation is non-homogeneous. If that f(x,t) hadn't been in the equation, you would have gotten L(0)=0 and the equation would have been homogeneous.