Lec 14 MIT 18.03 Differential Equations, Spring 2006
I just recalling some of the notation we are going to need for today, and a couple of the facts that we're going to use, plus trying to clear up a couple of confusions that the recitations report. This can be thought of two ways. It's a formal polynomial in D, in the letter D. It just has the shape of the polynomial, D squared plus AD plus B. A and B are constant coefficients. But, it's also, at the same time, if you think what it does, it's a linear operator on functions. It's a linear operator on functions like y of t. You think of it both ways: formal polynomial because we want to do things like factoring it, substituting two for D and things like that. Those are things you do with polynomials. You do them algebraically. You can take the formal derivative of the polynomial because it's just sums of powers. On the other hand, as a linear operator, it does something to functions. It differentiates them, multiplies them by constants or something li...