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

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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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