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Unformatted text preview: O. Linear Differential Operators 1. Linear differential equations. The general linear ODE of order n is (1) y ( n ) + p 1 ( x ) y ( n 1) + . . . + p n ( x ) y = q ( x ) . If q ( x ) = 0, the equation is inhomogeneous . We then call (2) y ( n ) + p 1 ( x ) y ( n 1) + . . . + p n ( x ) y = 0 . the associated homogeneous equation or the reduced equation. The theory of the n-th order linear ODE runs parallel to that of the second order equation. In particular, the general solution to the associated homogeneous equation (2) is called the complementary function or solution, and it has the form (3) y c = c 1 y 1 + . . . + c n y n , c i constants , where the y i are n solutions to (2) which are linearly independent , meaning that none of them can be expressed as a linear combination of the others, i.e., by a relation of the form (the left side could also be any of the other y i ): y n = a 1 y 1 + . . . + a n 1 y n 1 , a i constants. Once the associated homogeneous equation (2) has been solved by finding n independent solutions, the solution to the original ODE (1) can be expressed as (4) y = y p + y c , where y p is a particular solution to (1), and y c is as in (3). 2. Linear differential operators with constant coecients From now on we will consider only the case where (1) has constant coecients. This type of ODE can be written as (5) y ( n ) + a 1 y ( n 1) + . . . + a n y = q ( x ) ; using the differentiation operator D , we can write (5) in the form ( D n + a 1 D n 1 (6) + . . . + a n ) y = q ( x ) or more simply, p ( D ) y = q ( x ) , where y (7) p ( D ) = D n + a 1 D n 1 + . . . + a n . We call p ( D ) a polynomial differential operator with constant coecients . We think of the formal polynomial p ( D ) as operating on a function y ( x ), converting it into another function; it is like a black box, in which the function y ( x ) goes in, p(D)y and p ( D ) y (i.e., the left side of (5)) comes out. p(D) 1 2 18.03 NOTES Our main goal in this section of the Notes is to develop methods for finding particular solutions to the ODE (5) when q ( x ) has a special form: an exponential, sine or cosine, x k , or a product of these. (The function q ( x ) can also be a sum of such special functions.) These are the most important functions for the standard applications. The reason for introducing the polynomial operator p ( D ) is that this allows us to use polynomial algebra to help find the particular solutions. The rest of this chapter of the Notes will illustrate this. Throughout, we let (7) p ( D ) = D n + a 1 D n 1 + . . . + a n , a i constants . 3. Operator rules. Our work with these differential operators will be based on several rules they satisfy....
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This note was uploaded on 10/21/2011 for the course MATH 18.03 taught by Professor Vogan during the Fall '09 term at MIT.
- Fall '09