Linear Differential Operator

Section Linear Differential Operator Definition: I f L i s a l inear d ifferential op erator wi th co nstant co efficients and f is a sufficiently differentiable f unction s uch that L(f (x)) = 0, then L is said to be an annihilator of the function. Fact: The differential o perator D n a nnihilates e ach o f t he functions 1, x, x2 , . . . . . . .., xn−1 . Fact: The differential operator (D − α)n annihilates each of the functions eαx , xeαx , x2 eαx , . . . . . . , xn−1 eαx . Fact: The differential operator [D2 − 2αD + (α2 + β 2 )]n annihilates each of the functions eαx cos βx, xeαx cos βx, x2 eαx cos βx, ......., xn−1 eαx cos βx, eαx sin βx, xeαx sin βx x2 eαx sin βx,......., xn−1 eαx sin βx. Fact: Suppose that L1 and L2 are linear differential operators with constant coefficients such that L1 annihilates y1 (x) and L2 annihilates y2 (x), but L1 (y2 ) ̸= 0 and L2 (y1 ) ̸= 0. Then the product of differential operators L1 L2 annihilates the sum c1 y1 (x) + c2 y2 (x). Fact: The differential operator that annihilates a function is not unique. If a differential operator annihilates a function, then any other differential operator that has the operator as a factor also annihilates the function. Exercise 1. Write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L. (#2,4) (a) y ′′ − 5y = x2 − 2x. (b) 2y ′′ − 3y ′ − 2y = 1. Class Exercise 1. Write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L. (#6,8,10) (a) y ′′′ + 4y ′ = ex cos 2x (b) y ′′′ + 4y ′′ + 3y ′ = x2 cos x − 3x (4) ′′ 3 (c) y − 8y + 16y = (x − 2x)e4x Exercise 2. Verify that the differential operator 2D − 1 annihilates the function y = 4ex/2 . (#12) Class Exercise 2. Verify that the differential operator D2 + 64 annihilates the function y = 2 cos 8x − 5 sin 8x. (#14) Exercise 3. Find a differential operator that annihilates 6xe−4x + 5ex sin 2x. (Nagle Section 6.3 Example 1) Exercise 4. Find a linear differential operator that annihilates the given function. (#16, 18) (a) x3 (1 − 5x) (b) x + 3xe6x Class Exercise 3. Find a linear differential operator that annihilates the given function. (#20, 22, 24, 26) (a) 1 + sin x (b) 8x − sin x + 10 cos 5x (c) (2 − ex )2 (d) e−x sin x − e2x cos x Exercise 5. Find linearly independent functions that are annihilated by the given differential operator. (#28, 32) (a) D2 + 4D (b) D2 − 6D + 10 Class Exercise 4. Find linearly independent functions that are annihilated by the given differential operator. (#30, 34) (a) D2 − 9D − 36 (b) D2 (D − 5)(D − 7) Exercise 6. Find a general solution, using the annihilator method, to y ′′′ − 3y ′′ + 4y = xe2x . (Nagle Section 6.3 Example 3) Method of Undetermined Coefficients - Annihilator Approach (Page 158) The differential equation L(y) = g(x) has constant coefficients, and the function g(x) consists of finite sums and products of constants, polynomials, exponential functions, eαx , sines, and cosines. (i) Find the complementary solution yc for the homogeneous equation L(y) = 0. (ii) Operate on both sides of the nonhomogeneous equation L(y) = g(x) with a differential operator L1 that annihilates the function g(x). (iii) Find the general solution of the higher-order homogeneous differential equation L1 L(y) = 0. (iv) Delete from the solution in step (iii) all those terms that are duplicated in the complementary solution yc found in step (i). Form a linear combination yp of the terms that remain. This is the form of a particular solution of L(y) = g(x). (v) Substitute yp found in step (iv) into L(y) = g(x). Match coefficients of the various functions on each side of the inequality, and solve the resulting system of equations for the unknown coefficients in yp . (vi) With the particular solution found in step (v), form the general solution y = yc + yp of the given differential equation. Exercise 7. Solve the given differential equation by undetermined coefficients. (#36, 38, 40) (a) 2y ′′ − 7y ′ + 5y = −29 (b) y ′′′ + 2y ′′ + y ′ = 10 (c) y ′′ + 3y ′ = 4x − 5 Class Exercise 5. Solve the given differential equation by undetermined coefficients. (#42, 44, 46, 48, 50, 52, 54, 56, 58) (a) y ′′ − 2y ′ + y = x3 + 4x (b) y ′′ + 2y ′ + 2y = 5e6x ′′ ′ −2x (c) y + 6y + 8y = 3e + 2x (d) y ′′ + 4y = 4 cos x + 3 sin x − 8 ′′ ′ x (e) y + 3y − 10y = x(e + 1) (f) y ′′ + 2y ′ + y = x2 (e−x ) 1 x ′′ ′ (g) y + y + 4 y = e (sin 3x − cos 3x) (h) y ′′ + y = 4 cos x − sin x (i) y ′′ + 4y = cos2 x Exercise 8. Solve the given initial-value problem. y ′′ + y ′ = x, y(0) = 1, y ′ (0) = 0 (#66) Class Exercise 6. Solve the given initial-value problem. (#68, 70) (a) y ′′ + 5y ′ − 6y = 10e2x , y(0) = 1, y ′ (0) = 1 (b) y ′′′ − 2y ′′ + y ′ = xex + 5, y(0) = 2, y ′ (0) = 2, y ′′ (0) = -1 Homework: 3-71 (every 4th)