# 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.

### Constant Coefficients Method

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, ${y}_{h}\left(t\right)$$y_h(t)$, solution.

#### Investigation 05

Solve the following ${n}^{\text{th}}$$n^{\text{th}}$-order homogeneous ODEs using linear combination of $n$$n$ linearly-independent solutions:

1. ${y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }-6{y}^{\phantom{\rule{0.167em}{0ex}}\prime }+9y=0$$\displaystyle y^{\,\prime\prime}-6y^{\,\prime}+9y=0$

2. ${y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }+3{y}^{\phantom{\rule{0.167em}{0ex}}\prime }+2y=0$$\displaystyle y^{\,\prime\prime}+3y^{\,\prime}+2y=0$

3. ${y}^{\phantom{\rule{0.167em}{0ex}}\prime }=-4y$$\displaystyle y^{\,\prime}=-4y$

4. $3{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime \prime }-4{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }+{y}^{\phantom{\rule{0.167em}{0ex}}\prime }=0$$\displaystyle 3y^{\,\prime\prime\prime}-4y^{\,\prime\prime}+y^{\,\prime}=0$

5. ${y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime \prime }-5{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }-6{y}^{\phantom{\rule{0.167em}{0ex}}\prime }=0$$\displaystyle y^{\,\prime\prime\prime}-5y^{\,\prime\prime}-6y^{\,\prime}=0$

6. ${y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime \prime }-25{y}^{\phantom{\rule{0.167em}{0ex}}\prime }=0$$\displaystyle y^{\,\prime\prime\prime}-25y^{\,\prime}=0$

7. ${y}^{\phantom{\rule{0.167em}{0ex}}\left(5\right)}\left(t\right)-8{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime \prime }\left(t\right)+2{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }\left(t\right)+15{y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(t\right)-10y\left(t\right)=0$$\displaystyle y^{\,(5)}(t)-8y^{\,\prime\prime\prime}(t)+2y^{\,\prime\prime}(t)+15y^{\,\prime}(t)-10y(t)=0$

### Solve Polynomial Equations

#### Use Technology to Find Roots of Characteristic Equations

Mathematica

1(* Example: Solve x^4-4x^3-3x^2+18x=0 *)2Solve[x^4 - 4x^3 - 3x^2 + 18x == 0, x]

#### Use Technology to Verify Solutions to ${n}^{\text{th}}$$n^{\text{th}}$-order homogeneous ODEs

​x1(* Example: Solve y'''(x) - 3y''(x) - 40 y'(x) = 0 *)2(* DSolve uses function notation for dependent variable; and has 3 arguments *)3(* 1. ODE equation with == for = *)4(* 2. dependent variable, in function notation *)5(* 3. independent variable *)6
7DSolve[y'''[x] - 3y''[x] - 40 y'[x] == 0, y[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.