O. Linear Differential Operators 1. Linear differential equations. The general linear ODE of order n is (1) y(n) + p1(x)y(n−1) + . . . + pn(x)y = q(x). If q(x) �= 0, the equation is inhomogeneous. We then call (2) y(n) + p1(x)y(n−1) + . . . + pn(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) yc = c1y1 + . . . + cnyn , ci constants, where the yi 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 yi): yn = a1y1 + . . . + an−1yn−1 , ai 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 = yp + yc, where yp is a particular solution to (1), and yc is as in (3). 2. Linear differential operators with constant coefficients From now on we will consider only the case where (1) has constant coefficients. This type of ODE can be written as (5) y(n) + a1y(n−1) + . . . + any = q(x) ; using the differentiation operator D, we can write (5) in the form (Dn + a1Dn−1(6) + . . . + an) y = q(x) or more simply, p(D) y = q(x) , where y(7) p(D) = Dn + a1Dn−1 + . . . + an . We call p(D) a polynomial differential operator with constant coefficients. 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)yand p(D)y (i.e., the left side of (5)) comes out. p(D) 12 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, xk , 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) = Dn + a1Dn−1 + . . . + an , ai constants. 3. Operator rules. Our work with these differential operators will be based on several rules they satisfy. In stating these rules, we will always assume that the functions involved are sufficiently differentiable, so that the operators can be applied to them. Sum rule. If p(D) and q(D) are polynomial operators, then for any (sufficiently differ-entiable) function u, (8) [p(D) + q(D)]u = p(D)u + q(D)u . Linearity rule. If u1 and u2 are functions, and ci constants, (9) p(D) (c1u1 + c2u2) = c1p(D) u1 + c2p(D) u2 . The linearity rule is a familiar property of the operator a Dk ; it extends to sums of these operators, using the sum rule above, thus it is true for operators which are polynomials in D. (It is still true if the coefficients ai in (7) are not constant, but functions of x.) Multiplication rule. If p(D) = g(D) h(D), as polynomials in D, then u (10) p(D) u = g(D)�h(D) u . The picture illustrates the meaning of the right side of (10). The property h(D)u is true when h(D) is the simple operator a Dk , essentially because Dm(a Dk u) = a Dm+k u; it extends to general polynomial operators h(D) by linearity. Note that a must be a constant; it’s false otherwise. p(D)u g(D) h(D) An important corollary of the multiplication property is that polynomial operators with constant coefficients commute; i.e., for every function u(x), (11) g(D)�h(D) u = h(D)�g(D) u . For as polynomials, g(D)h(D) = h(D)g(D) = p(D), say; therefore by the multiplication rule, both sides of (11) are equal to p(D) u, and therefore equal to each other. The remaining two rules are of a different type, and more concrete: they tell us how polynomial operators behave when applied to exponential functions and products involving exponential functions. Substitution rule.3 O. LINEAR DIFFERENTIAL OPERATORS ax(12) p(D)e ax = p(a)e Proof. We have, by repeated differentiation, ax ax 2 ax ax k axDe ax = ae , D2 e = a e , . . . , Dk e = a e ; therefore, ax ax(Dn + c1Dn−1 + . . . + cn) e = (a n + c1a n−1 + . . . + cn) e , which is the substitution rule (12). � The exponential-shift rule This handles expressions such as xk eax and xk sin ax. ax(13) p(D) e ax u = e p(D + a) u . Proof. We prove it in successive stages. First, it is true when p(D) = D, since by the product rule for differentiation, ax(14) De ax u(x) = e axDu(x) + ae u(x) = e ax(D + a)u(x) . To show the rule is true for Dk , we apply (14) to D repeatedly: ax D2 e u = D(De ax u) = D(e ax(D + a)u) by (14); = e ax(D + a)�(D + a)u, by (14); ax= e (D + a)2 u , by (10). In the same way, ax axD3 e u = D(D2 e u) = D(e ax(D + a)2 u) by the above; = e ax(D + a)�(D + a)2 u, by (14); ax= e (D + a)3 u , by (10), and so on. This shows that (13) is true for an operator of the form Dk . To show it is true for a general operator p(D) = Dn + a1Dn−1 + . . . + an , we write (13) for each Dk (eaxu), multiply both sides by the coefficient ak, and add up the resulting equations for the different values of k. � Remark on complex numbers. By Notes C. (20), the formula ax(*) D (c e ax) = c a e remains true even when c and a are complex numbers; therefore the rules and arguments above remain valid even when the exponents and coefficients are complex. We illustrate. Example 1. Find D3e−x sin x . Solution using the exponential-shift rule. Using (13) and the binomial theorem, −x −x(D3D3 e −x sin x = e (D − 1)3 sin x = e − 3D2 + 3D − 1) sin x −x= e (2 cos x + 2 sin x), since D2 sin x = − sin x, and D3 sin x = − cos x. Solution using the substitution rule. Write e−x sin x = Im e(−1+i)x . We have D3 (−1+i)x e = (−1 …
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