1) I have been asked to classify some differential equations as boundary or initial problems. With a first order ordinary DE does this even make sense? There is only one condition to be met y(a) = b
2) If the dependent variable is subject to a modulus operation in one of its terms, does this make the differential equation non-linear?
1) This is not a time equation necessarily. In second order equations we still have an initial condition problem if y(a) = b and y'(a) = c as I understand it... Does your position remain the same?
2) I understand what the modulus function implies about the value of y. My question is "does this classify as non-linear behavior?" I Suspect yes. It's the definition of a linear equation that I require clarification of, rather than the modulus function.
Like Mr F said, it is called an "initial condition" when the value you are given of your independent variable is . This is simply because intuitively this is where we interpret the independent variable as starting. So it doesn't matter if the independent variable is time or not.