Homogeneous Equations

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

• Section 4.2 – Homogeneous Linear Equations: The General Solution

Expected Educational Results

• Objective 10–1: I can identify if two or more functions are linearly-independent.

• Objective 10–2: I can identify the characteristic equation for $n^{\text{th}}$$n^{\text{th}}$-degree homogeneous linear ODEs.

• Objective 10–3: I can find the most general solution to $n^{\text{th}}$$n^{\text{th}}$-degree homogeneous linear ODEs.

Linear Independence

Definition: Linear Independence of Two Functions

Two functions $y_1(t)$$y_1(t)$ and $y_2(t)$$y_2(t)$ are linearly independent on some interval if and only if neither of the functions is a constant multiple of the other on the interval.

Definition: Linear Dependence of Two Functions

Two functions $y_1(t)$$y_1(t)$ and $y_2(t)$$y_2(t)$ are linearly dependent on some interval if and only if one of the functions is a constant multiple of the other on the interval.

Definition: Wronskian

The Wronskian of $y_1$$y_1$ and $y_2$$y_2$, ${\displaystyle W[y_1,y_2,\dots ,y_n](t)}$$\displaystyle W[y_1,y_2,\ldots ,y_n](t)$, is the determinant of the $n\times n$$n\times n$ matrix:

$\begin{array}{c}\left(\begin{array}{cccc}{y}_1(\tau )& y_2(\tau )& \cdots & y_n(\tau )\\ y_1^{\prime }(\tau )& y_2^{\prime }(\tau )& \cdots & y_n^{\prime }(\tau )\\ ⋮& ⋮& ⋮& ⋮\\ y_1^{(n)}(\tau )& y_2^{(n)}(\tau )& \cdots & y_n^{(n)}(\tau )\end{array}\right)\end{array}$$\left( \begin{array}{cccc} y_1(\tau) & y_2(\tau) & \cdots & y_n(\tau) \\ y_1^{\,\prime}(\tau) & y_2^{\,\prime}(\tau) & \cdots & y_n^{\,\prime}(\tau) \\ \vdots & \vdots & \vdots & \vdots \\ y_1^{\,(n)}(\tau) & y_2^{\,(n)}(\tau) & \cdots & y_n^{\,(n)}(\tau) \\ \end{array} \right)$

The Wronskian can be used to determine the linear independence of $n$$n$ functions.

Theorem: Linear Independence

If ${\displaystyle W[y_1,y_2,\dots ,y_n](t)\ne 0}$$\displaystyle W[y_1,y_2,\ldots,y_n](t)\ne 0$, then $y_1$$y_1$, $y_2$$y_2$, $\dots$$\ldots$, $y_n$$y_n$ are linearly independent.

Investigation 01

Use the Wronskian to determine if the following list of functions are linearly independent:

1. ${\displaystyle y=-x^2}$$\displaystyle y=-x^2$, ${\displaystyle y=1+4x^2}$$\displaystyle y=1+4x^2$

2. ${\displaystyle y=e^{3x}}$$\displaystyle y=e^{3x}$, ${\displaystyle y=x{e}^{3x}}$$\displaystyle y=xe^{3x}$

3. ${\displaystyle y=\cos (x)}$$\displaystyle y=\cos{(x)}$, ${\displaystyle y=\sin (x)}$$\displaystyle y=\sin{(x)}$

4. ${\displaystyle y=\cos (x)}$$\displaystyle y=\cos{(x)}$, ${\displaystyle y=x\cos (x)}$$\displaystyle y=x\cos{(x)}$

5. ${\displaystyle y=x}$$\displaystyle y=x$, ${\displaystyle y=\cos (x)}$$\displaystyle y=\cos{(x)}$

6. ${\displaystyle y={\cos }^2(x)}$$\displaystyle y=\cos^2{(x)}$, ${\displaystyle y={\sin }^2(x)}$$\displaystyle y=\sin^2{(x)}$

7. ${\displaystyle y=e^x}$$\displaystyle y=e^x$, ${\displaystyle y=e^{2x}}$$\displaystyle y=e^{2x}$, ${\displaystyle y=e^{3x}}$$\displaystyle y=e^{3x}$

8. ${\displaystyle y=e^{-x}}$$\displaystyle y=e^{-x}$, ${\displaystyle y=x{e}^{-x}}$$\displaystyle y=xe^{-x}$, ${\displaystyle y=x^2{e}^{-x}}$$\displaystyle y=x^2e^{-x}$

9. ${\displaystyle y=3x^2+3x+8}$$\displaystyle y=3x^2+3x+8$, ${\displaystyle y=2x^2+2x+2}$$\displaystyle y=2x^2+2x+2$, ${\displaystyle y=5x^2-2x+8}$$\displaystyle y=5x^2-2x+8$

Computing the Wronskian

Use Technology to Compute the Wronskian

Mathematica

​x
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(* Example from Investigation 01 *)
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(* Compute the Wronskian for y=cos(x) and y=x*cos(x) *)
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(* The functions are grouped inside braces; separated by comma *)
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(* The second argument is the independent variable *)
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Wronskian[{Cos[x], x Cos[x]}, x]

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.

2. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Last Modified: Monday, 6 September 2020 17:33 EDT