# Differential Operators and the Elimination Method for Systems

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 5.2 – Differential Operators and the Elimination Method for Systems

## Expected Educational Results

• Objective 15–1: I can identify the linear differential operator for any polynomial function, exponential function, or sine and cosine functions.
• Objective 15–2: I can solve linear systems of ODEs using linear differential operators.

## Differential Operators

### Properties of Linear Differential Operators

Let $p(D)$, $q(D)$, $g(D)$, $h(D)$ be polynomial differential operators; $u_1$ and $u_2$ be sufficiently differentiable functions of $x$; $c_1$ and $c_2$ constants.

• Sum Rule: $\left(p(D)+q(D)\right)[u]=p(D)[u]+q(D)[u]$
• Product: $D[D[u]]=D^2[u]$.
• Linearity Rule: $p(D)\left[c_1u_1+c_2u_2\right]=c_1p(D)[u_1]+c_2p(D)[u_2]$
• Multiplication Rule: If $p(D)=g(D)h(D)$, then $p(D)[u]=g(D)\left[h(D)[u]\right]$
• Commutativity of Multiplication: If $g(D)$ and $h(D)$ have constant coefficients, then $g(D)h(D)[u]=h(D)g(D)[u]$.
• Substitution Rule: $p(D)[e^{ax}]=p(a)e^{ax}$
• Exponential-Shift Rule: $p(D)[e^{ax}u]=e^{ax}p(D+a)[u]$

#### Investigation 10

Prove any property of linear differential operators.

#### Investigation 11

1. Show $(D+2)(D-1)$ is equivalent to $D^2+D-2$.
2. Show $(D-3)(D+1)$ is equivalent to $(D+1)(D-3)$.
3. Show $(D-3t)(D+1)$ is not equivalent to $(D+1)(D-3t)$.

### Annihilators

#### Definition: Annihilator, $A(D)$

A linear differential operator, $A(D)$ is said to annihilate a function $f(x)$ if $A(D)\left[f(x)\right]=0$ for all $x$.

#### Forms of Annihilators

• For polynomials of degree $n$, $A(D)=D^{\,n+1}$
• For $e^{\alpha x}$, $A(D)=D-\alpha$
• For $t^{\,n}e^{\alpha x}$, $A(D)=(D-\alpha)^{\,n+1}$
• For $\sin{(\beta x)}$ or $\cos{(\beta x)}$, $A(D)=D^2+\beta^2$
• For $t^{\,n}e^{\alpha x}\sin{(\beta x)}$, $A(D)=\left[(D-\alpha)^2+\beta^2\right]^{\,n+1}$

Note: You do not to memorize all of the above $A(D)$. How can you succinctly remember the forms for each $A(D)$?