Homogeneous Equations

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

Expected Educational Results

Homogeneous Linear DEs

Definition: Second-Order Homogeneous Linear Equations

ay+by+cy=0, a0, a,b,cR.

Definition: nth-Order Homogeneous Linear Equations

any(n)+an1y(n1)++a2y+a1y+a0y=0, an0, aiR.

Definition: Characteristic Equation

For the homogeneous linear equation shown in Investigation 02, the characteristic equation is ar2+br+c=0, where r1 and r2 are the roots of the characteristic equation.

Definition: General Solution

If y1 and y2 are linearly independent solutions to ay+by+cy=0, a0 on (,), then the general solution to the homogeneous linear DE is a linear combination of y1 and y2:

yg(t)=c1y1(t)+c2y2(t)

Investigation 03

Let y1(t) and y2(t) be linearly-independent solutions to ay(t)+by(t)+cy(t)=0. Show that c1y1(t)+c2y2(t) is also a solution to ay(t)+by(t)+cy(t)=0.

Solutions to Linear Homogeneous ODEs

There are three possibilities for general solutions [corresponding to the roots of the characteristic equation]:

  1. If r1 and r2 are distinct real roots of the characteristic equation, then the general solution is yg(t)=c1er1t+c2er2t;

  2. r1 is a repeated root of the characteristic equation, then the general solution is yg(t)=c1er1t+c2ter1t;

  3. r1 and r2 are complex conjugate root of the characteristic equation, then we will discuss this case next section.

Investigation 04

Suppose the following are roots to the characteristic equation. Find the homogeneous solution, yh(x).

  1. r=2,1

  2. r=0,2,3

  3. r=1,1,1

  4. r=3,0,0,2

  5. r=1,1,1,2,3

  6. r=2,2,2

  7. r=1+3,13

Existence and Uniqueness

Theorem: Existence and Uniqueness Theorem

For any real numbers a (0), b, c, t0, Y0, and Y1, there exists a unique solution to the IVP ay+by+cy=0 on (,), y(t0)=Y0, y(t0)=Y1 and the solution is valid for all t in (,).

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