To be linear, .
How is this equation (see below) linear?
You have to distinguish between linear operators and linear equations. Typically, for the equation you've given, you'd throw the 3 over to the RHS and consider the operator to be everything else (that is, all the stuff with the dependent variable in it). So the operator is
This is linear. So a differential equation is considered linear if the associated operator is linear. The DE can still be linear even if it's non-homogeneous, as in your case. Does that make sense?