Differential Operators; Laws of Operation MATH 374 Lecture 13 Differential Operators; Laws of Operation

4.7: Differential Operators We will now define some notation to make working with differential equations easier! Definition: 2

Applying an Operator to a Function If we apply operator A as defined in (1) to a function y with sufficient derivatives, we get Note that Boyce and Diprima use L for differential operators and write L[y]. 3

Equal Operators; Product of two Operators Definition: Two operators A and B are said to be equal if and only if Ay = By for all functions y for which A and B make sense. Definition: The product of two operators A and B is defined by ABy := A(By) Note that AB is itself an operator. 4

Differential Operators are Linear Remark: Differential operators are linear, i.e. A(c1 f1 + c2 f2) = c1Af1 + c2 Af2 for all constants c1 and c2 and all functions f1 and f2 for which A makes sense. 5

Example 1: Let A = 4D+3 and B = D-6. Find AB and BA. Solution: Apply AB and BA to an arbitrary function y: Find AB: By = (D-6)y = Dy – 6y = y’ – 6y A(By) = A(y’ – 6y) = Ay’ – 6Ay = (4D+3)y’ – 6(4D+3)y = 4Dy’+3y’ – 6(4Dy+3y) = 4y’’ + 3y’ – 24y’ – 18y = 4y’’ – 21y’ – 18y = (4D2 – 21D – 18)y Therefore AB = 4D2 – 21D – 18 Find BA: Ay = (4D + 3)y = 4Dy + 3y = 4y’ + 3y B(Ay) = B(4y’ + 3y) = 4By’ + 3By = 4(D – 6)y’ + 3(D – 6)y = 4(Dy’ – 6y’) +3(Dy – 6y) = 4y’’ – 24y’ + 3y’ – 18y = 4y’’ – 21y’ – 18y = (4D2 – 21D – 18)y Therefore BA = 4D2 – 21D – 18. Note in this case that AB = BA! 6

Example 2: Let A = x2 D and B = D+2. Find AB and BA. Solution: (AB)y = A(By) = A(D+2)y = A(Dy + 2y) = A(y’+2y) = Ay’ + 2Ay = (x2D)y’+ 2(x2D)y = x2 y’’ + 2x2 y’ =(x2D2+2x2D)y Therefore AB = x2D2+2x2D. (BA)y = B(Ay) = B(x2Dy) = B(x2 y’) = (D+2)(x2 y’) = D(x2 y’) + 2(x2 y’) = (2xy’ + x2y’’) + 2x2y’ = x2y’’ + (2x2 + 2x)y’ = (x2D2 + (2x2 + 2x)D)y Therefore BA = x2D2 + (2x2 + 2x)D Note that in this case, AB  BA. 7

Constant Coefficients; Sum of two Differential Operators Remark: In general, differential operators do not commute. If A and B have constant coefficients, then AB = BA. Definition: If A = a0Dn + … + an-1D + an and B = b0Dn + … + bn-1D + bn, then the sum of A and B is the operator (A+B):= (a0+b0)Dn + … + (an-1+bn-1)D + (an+bn). 8

4.8: Fundamental Laws of Operations for Differential Operators For any differential operators A, B, and C, A+B = B+A (Commutative Law of Addition) (A+B)+C = A+(B+C) (Associative Law of Addition) (AB)C = A(BC) (Associative Law of Multiplication) A(B+C) = AB + AC (Distributive Law of Multiplication over Addition) If A and B have constant coefficients, then AB = BA. (Commutative Law of Multiplication) For any positive integers m and n, Dm Dn = Dm+n. (Law of Exponents) Note: Operators with constant coefficients behave like algebraic polynomials! 9

Example 1 If A = 4D+3 and B = D-6, then AB = (4D+3)(D-6) = 4D2 – 24D + 3D – 18 = 4D2 – 21D – 18 (Compare to the first example in the last section!) 10