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

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

#### Investigation 06

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

1. ${y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }+4{y}^{\phantom{\rule{0.167em}{0ex}}\prime }+3y=0$$\displaystyle y^{\,\prime\prime}+4y^{\,\prime}+3y=0$; $y\left(0\right)=0$$y(0)=0$ and ${y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(0\right)=20$$y^{\,\prime}(0) = 20$

2. ${y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }+6{y}^{\phantom{\rule{0.167em}{0ex}}\prime }+9y=0$$\displaystyle y^{\,\prime\prime}+6y^{\,\prime}+9y=0$; $y\left(0\right)=1$$y(0)=1$ and ${y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(0\right)=2$$y^{\,\prime}(0) = 2$

3. ${y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime \prime }\left(t\right)+{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }\left(t\right)-6{y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(t\right)=0$$\displaystyle y^{\,\prime\prime\prime}(t)+y^{\,\prime\prime}(t)-6y^{\,\prime}(t)=0$, $y\left(0\right)=2$$y(0)=2$, ${y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(0\right)=-3$$y^{\,\prime}(0)=-3$, ${y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }\left(0\right)=1$$y^{\,\prime\prime}(0)=1$

Example 01: Solve ${y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime \prime }\left(x\right)-3{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }\left(x\right)-40{y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)=0$$y^{\,\prime\prime\prime}(x) - 3y^{\,\prime\prime}(x) - 40 y^{\,\prime}(x) = 0$, where $y\left(0\right)=0$$y(0)=0$, ${y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(0\right)=-2$$y^{\,\prime}(0)=-2$, ${y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }\left(0\right)=1$$y^{\,\prime\prime}(0)=1$.

Solution:

The characteristic equation is ${r}^{3}-3{r}^{2}-40r=0⇒r=-5,0,8$$r^3-3r^2-40r=0\Rightarrow r=-5, 0, 8$.

#### Use Technology to Solve Characteristic Equation

1(* Example: Solve y'''(x) - 3y''(x) - 40 y'(x) = 0 *)2(* Solve characteristic equation *)3Solve[r^3 - 3r^2 - 40r == 0, r]

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, ${y}_{h}\left(x\right)={c}_{1}{e}^{-5x}+{c}_{2}+{c}_{3}{e}^{8x}$$y_h(x)=c_1e^{-5x}+c_2+c_3e^{8x}$.

Solve for the constants in the IVP:

${y}_{h}\left(0\right)={c}_{1}{e}^{-5\left(0\right)}+{c}_{2}+{c}_{3}{e}^{8\left(0\right)}=0⇒{c}_{1}+{c}_{2}+{c}_{3}=0$$y_h(0)=c_1e^{-5(0)}+c_2+c_3e^{8(0)}=0\Rightarrow c_1+c_2+c_3=0$

${y}_{h}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(0\right)=-5{c}_{1}{e}^{-5\left(0\right)}+8{c}_{3}{e}^{8\left(0\right)}=-2⇒-5{c}_{1}+8{c}_{3}=-2$$y_h^{\,\prime}(0)=-5c_1e^{-5(0)}+8c_3e^{8(0)}=-2\Rightarrow -5c_1+8c_3=-2$

${y}_{h}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }\left(0\right)=25{c}_{1}{e}^{-5\left(0\right)}+64{c}_{3}{e}^{8\left(0\right)}=1⇒25{c}_{1}+64{c}_{3}=1$$y_h^{\,\prime\prime}(0)=25c_1e^{-5(0)}+64c_3e^{8(0)}=1\Rightarrow 25c_1+64c_3=1$

Solve the following system of equations for ${c}_{1}$$c_1$, ${c}_{2}$$c_2$, ${c}_{3}$$c_3$:

${c}_{1}+{c}_{2}+{c}_{3}=0$$c_1+c_2+c_3=0$ $-5{c}_{1}+8{c}_{3}=-2$$-5c_1+8c_3=-2$ $25{c}_{1}+64{c}_{3}=1$$25c_1+64c_3=1$

#### Use Technology to Solve for Constants in ${n}^{\text{th}}$$n^{\text{th}}$-order Homogeneous IVPs

​x1(* Example: Solve y'''(x) - 3y''(x) - 40 y'(x) = 0 *)2(* Solve for constants *)3(* Dolve has 2 arguments *)4(* 1. all equations with grouped with braces *)5(* 2. all constants grouped with braces *)6
7Solve[{c1 + c2 + c3 == 0, -5c1 + 8c3 == -2, 25c1 + 64c3 == 1}, {c1, c2, c3}]

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 ${n}^{\text{th}}$$n^{\text{th}}$-order Homogeneous IVPs

xxxxxxxxxx91(* Example: Solve y'''(x) - 3y''(x) - 40 y'(x) = 0 *)2(* where y[0]==0, y'[0]==-2, y''[0]==1 *)3
4(* DSolve uses function notation for dependent variable; and has 3 arguments *)5(* 1. ODE equation with == for =; grouped with initial conditions *)6(* 2. dependent variable, in function notation *)7(* 3. independent variable *)8
9DSolve[{y'''[x] - 3y''[x] - 40 y'[x] == 0, y[0]==0, y'[0]==-2, y''[0]==1}, 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.