Prediction with Optional Features

Here's a question/scenario that has me going "hmm" ... I am faced with a regression problem where my dataset has examples with attributes {A, B, C} and other examples have attributes {A, B, C, D, E}. I'm scratching my head as I consider different ways to model the data to ultimately predict the target variable.

I understand at a basic level that my regression formula can't be Y = f(A,B,C,D,E) unless I have a way to impute/default the values "D" and "E" for those examples without those features. My thought process is "my model can make a more accurate prediction when it has more information" which is the hypothesis I want to prove with this data.

Anybody have experience developing a model(s) when some of the attributes are "optional?"