Homogeneous Equations

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

Expected Educational Results

Constant Coefficients Method for IVPs

  1. Identify the characteristic equation for the homogeneous linear DE.

  2. Completely solve the characteristic equation.

  3. Use the roots of the characteristic equation for the homogeneous, yh(t), solution.

  4. Use system of equations to solve for the constants.

Investigation 06

Solve the following nth-order IVPs using linear combination of n linearly-independent solutions:

  1. y+4y+3y=0; y(0)=0 and y(0)=20

  2. y+6y+9y=0; y(0)=1 and y(0)=2

  3. y(t)+y(t)6y(t)=0, y(0)=2, y(0)=3, y(0)=1

Example 01: Solve y(x)3y(x)40y(x)=0, where y(0)=0, y(0)=2, y(0)=1.

Solution:

The characteristic equation is r33r240r=0r=5,0,8.

Use Technology to Solve Characteristic Equation

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.

Thus, yh(x)=c1e5x+c2+c3e8x.

Solve for the constants in the IVP:

yh(0)=c1e5(0)+c2+c3e8(0)=0c1+c2+c3=0

yh(0)=5c1e5(0)+8c3e8(0)=25c1+8c3=2

yh(0)=25c1e5(0)+64c3e8(0)=125c1+64c3=1

Solve the following system of equations for c1, c2, c3:

c1+c2+c3=0 5c1+8c3=2 25c1+64c3=1

Use Technology to Solve for Constants in nth-order Homogeneous IVPs

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.

Use Technology to Verify Solutions to nth-order Homogeneous IVPs

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