Homogeneous Equations

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

Expected Educational Results

Linear Independence

Definition: Linear Independence of Two Functions

Two functions y1(t) and y2(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 y1(t) and y2(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 y1 and y2, W[y1,y2,,yn](t), is the determinant of the n×n matrix:

(y1(τ)y2(τ)yn(τ)y1(τ)y2(τ)yn(τ)y1(n)(τ)y2(n)(τ)yn(n)(τ))

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

Theorem: Linear Independence

If W[y1,y2,,yn](t)0, then y1, y2, , yn are linearly independent.

Investigation 01

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

  1. y=x2, y=1+4x2

  2. y=e3x, y=xe3x

  3. y=cos(x), y=sin(x)

  4. y=cos(x), y=xcos(x)

  5. y=x, y=cos(x)

  6. y=cos2(x), y=sin2(x)

  7. y=ex, y=e2x, y=e3x

  8. y=ex, y=xex, y=x2ex

  9. y=3x2+3x+8, y=2x2+2x+2, y=5x22x+8

Computing the Wronskian

Use Technology to Compute the Wronskian

Mathematica

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.

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