# 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

### Definition: Differential Operator

A differential operator is in the form, $D^{\,n}$, where $D^{\,n}$ is the $n^{\text{th}}$ derivative for $n\in\mathbb{W}$, i.e., $n=0,1,2,3,\ldots$.

NOTE: The differential operator always has a function as its argument. For example, $D^2[\sin{(2x)}]$ is the representation of the second derivative of $\sin{(2x)}$.

### Definition: Linear Differential Operators, $L(D)$

A linear operator of order $n$ is

$L(D)=aD^{\,n}+bD^{n-1}+cD^{n-2}+\cdots$ for $a,b,c,\ldots\in\mathbb{R}$

i.e., a Linear Differential Operator, $L(D)$ is a linear combination of differential operators.

#### Examples of Differential Operators

Example 01: $3D[6t^3]=3(18t^2)=54t^2$

Example 02: $(D^2-5)[\cos{(2t)}]=D^2[\cos{(2t)}]-5[\cos{(2t)}]=-4\cos{(2t)}-5\cos{(2t)}=-9\cos{(2t)}$

Example 03: $D(D+2)[te^{-t}]=(D^2+2D)[te^{-t}]=D^2[te^{-t}]+2D[te^{-t}]=-2e^{-t}+te^{-t}+2(e^{-t}-te^{-t})=-te^{-t}$

### Identifying Linear Differential Operators, $L(D)$

#### Investigation 01

Identify the lowest order $L(D)$ for the following functions, so that $L(D)\left[f(x)\right]=0$. Explain.

1. $f(x)=x$
2. $f(x)=x^2$
3. $f(x)=x^3$
4. $f(x)=x^n$, $n\geq 0$, $n\in\mathbb{W}$

#### Investigation 02

Show the lowest order linear differential operator, $L(D)$, for an exponential function, $e^{\alpha x}$, is $(D-\alpha)$, i.e., $(D-\alpha)\left[e^{\alpha x}\right]=0$. Explain.

1. $f(x)=e^{x}$
2. $f(x)=e^{2x}$
3. $f(x)=e^{-3x}$
4. $f(x)=e^{\frac{1}{2}x}$
5. $f(x)=e^{\alpha x}$

#### Investigation 03

Show the lowest order linear differential operator, $L(D)$, for the product of a polynomial function and exponential function, $x^n e^{\alpha x}$, is $(D-\alpha)^{n+1}$, i.e., $(D-\alpha)^{n+1}\left[x^n e^{\alpha x}\right]=0$. Explain.

1. $f(x)=xe^{x}$
2. $f(x)=x^2e^{x}$
3. $f(x)=x^2e^{2x}$
4. $f(x)=x^ne^{\alpha x}$, $n\geq 0$, $n\in\mathbb{W}$

#### Investigation 04

Show the lowest order linear differential operator, $L(D)$, for either $\cos{(\beta x)}$ or $\sin{(\beta x)}$, is $(D^2+\beta^2)$, i.e., $(D^2+\beta^2)\left[\cos{(\beta x)}\right]=0$ and $(D^2+\beta^2)\left[\sin{(\beta x)}\right]=0$. Explain.

1. $f(x)=\cos{(x)}$
2. $f(x)=\sin{(x)}$
3. $f(x)=\cos{(2x)}$
4. $f(x)=\cos{(\beta x)}$

#### Investigation 05

Show the lowest order linear differential operator, $L(D)$, for either $x^n\cos{(\beta x)}$ or $x^n\sin{(\beta x)}$, is $(D^2+\beta^2)^{n+1}$, i.e., $(D^2+\beta^2)^{n+1}\left[\cos{(\beta x)}\right]=0$ and $(D^2+\beta^2)^{n+1}\left[\sin{(\beta x)}\right]=0$. Explain.

1. $f(x)=x\cos{(x)}$
2. $f(x)=x^2\sin{(x)}$
3. $f(x)=x\cos{(2x)}$
4. $f(x)=x^n\cos{(\beta x)}$

#### Investigation 06

Show the lowest order linear differential operator, $L(D)$, for either $e^{\alpha x}\cos{(\beta x)}$ or $e^{\alpha x}\sin{(\beta x)}$, is

$((D-\alpha)^2+\beta^2)$, i.e., $((D-\alpha)^2+\beta^2)\left[e^{\alpha x}\cos{(\beta x)}\right]=0$ and $((D-\alpha)^2+\beta^2)\left[e^{\alpha x}\sin{(\beta x)}\right]=0$

Explain.

1. $f(x)=e^x\cos{(3x)}$
2. $f(x)=e^{2x}\sin{(x)}$
3. $f(x)=e^{-x}\cos{(2x)}$
4. $f(x)=e^{\alpha x}\cos{(\beta x)}$

#### Investigation 07

Show the lowest order linear differential operator, $L(D)$, for either $x^ne^{\alpha x}\cos{(\beta x)}$ or $x^ne^{\alpha x}\sin{(\beta x)}$, is $((D-\alpha)^2+\beta^2)^{n+1}$, i.e., $((D-\alpha)^2+\beta^2)^{n+1}\left[x^ne^{\alpha x}\cos{(\beta x)}\right]=0$ and $((D-\alpha)^2+\beta^2)^{n+1}\left[x^ne^{\alpha x}\sin{(\beta x)}\right]=0$. Explain.

1. $f(x)=x^2e^x\cos{(x)}$
2. $f(x)=xe^{2x}\sin{(3x)}$

#### Investigation 08

Prove the following:

1. $(D-2)[(D^2+9)[e^{2 x}+\cos{(3x)}]]=0$
2. $((D-2)^2+9)[e^{2 x}\cos{(3x)}]=0$

#### Investigation 09

Using your work from above, identify $L(D)$ for $x^ne^{\alpha x}\cos{(\beta x)}$, $n\geq 0$, $n\in\mathbb{W}$, and $x^ne^{\alpha x}\sin{(\beta x)}$, $n\geq 0$, $n\in\mathbb{W}$.