# 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.

### Homogeneous Linear DEs

#### Definition: Second-Order Homogeneous Linear Equations

$a{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }+b{y}^{\phantom{\rule{0.167em}{0ex}}\prime }+cy=0$$\displaystyle ay^{\,\prime\prime}+by^{\,\prime}+cy=0$, $a\ne 0$$a\ne 0$, $a,b,c\in \mathbb{R}$$a,b,c\in\mathbb{R}$.

#### Definition: ${n}^{\text{th}}$$n^{\text{th}}$-Order Homogeneous Linear Equations

${a}_{n}{y}^{\phantom{\rule{0.167em}{0ex}}\left(n\right)}+{a}_{n-1}{y}^{\phantom{\rule{0.167em}{0ex}}\left(n-1\right)}+\cdots +{a}_{2}{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }+{a}_{1}{y}^{\phantom{\rule{0.167em}{0ex}}\prime }+{a}_{0}y=0$$\displaystyle a_ny^{\,(n)}+a_{n-1}y^{\,(n-1)}+\cdots +a_2y^{\,\prime\prime}+a_1y^{\,\prime}+a_0y=0$, ${a}_{n}\ne 0$$a_n\ne 0$, ${a}_{i}\in \mathbb{R}$$a_i\in\mathbb{R}$.

#### Definition: Characteristic Equation

For the homogeneous linear equation shown in Investigation 02, the characteristic equation is $a{r}^{2}+br+c=0$$\displaystyle ar^2+br+c=0$, where ${r}_{1}$$r_1$ and ${r}_{2}$$r_2$ are the roots of the characteristic equation.

#### Definition: General Solution

If ${y}_{1}$$y_1$ and ${y}_{2}$$y_2$ are linearly independent solutions to $a{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }+b{y}^{\phantom{\rule{0.167em}{0ex}}\prime }+cy=0$$\displaystyle ay^{\,\prime\prime}+by^{\,\prime}+cy=0$, $a\ne 0$$a\ne 0$ on $\left(-\mathrm{\infty },\mathrm{\infty }\right)$$(-\infty,\infty)$, then the general solution to the homogeneous linear DE is a linear combination of ${y}_{1}$$y_1$ and ${y}_{2}$$y_2$:

${y}_{g}\left(t\right)={c}_{1}{y}_{1}\left(t\right)+{c}_{2}{y}_{2}\left(t\right)$$y_g(t)=c_1y_1(t)+c_2y_2(t)$

#### Investigation 03

Let ${y}_{1}\left(t\right)$$y_1(t)$ and ${y}_{2}\left(t\right)$$y_2(t)$ be linearly-independent solutions to $a{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }\left(t\right)+b{y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(t\right)+cy\left(t\right)=0$$ay^{\,\prime\prime}(t)+by^{\,\prime}(t)+cy(t)=0$. Show that ${c}_{1}{y}_{1}\left(t\right)+{c}_{2}{y}_{2}\left(t\right)$$c_1y_1(t)+c_2y_2(t)$ is also a solution to $a{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }\left(t\right)+b{y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(t\right)+cy\left(t\right)=0$$ay^{\,\prime\prime}(t)+by^{\,\prime}(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 ${r}_{1}$$r_1$ and ${r}_{2}$$r_2$ are distinct real roots of the characteristic equation, then the general solution is ${y}_{g}\left(t\right)={c}_{1}{e}^{{r}_{1}t}+{c}_{2}{e}^{{r}_{2}t}$$y_g(t)=c_1e^{r_1t}+c_2e^{r_2t}$;

2. ${r}_{1}$$r_1$ is a repeated root of the characteristic equation, then the general solution is ${y}_{g}\left(t\right)={c}_{1}{e}^{{r}_{1}t}+{c}_{2}t{e}^{{r}_{1}t}$$y_g(t)=c_1e^{r_1t}+c_2te^{r_1t}$;

3. ${r}_{1}$$r_1$ and ${r}_{2}$$r_2$ 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, ${y}_{h}\left(x\right)$$y_h(x)$.

1. $r=-2,1$$\displaystyle r=-2,1$

2. $r=0,2,3$$\displaystyle r=0,2,3$

3. $r=-1,-1,1$$\displaystyle r=-1,-1,1$

4. $r=-3,0,0,2$$\displaystyle r=-3,0,0,2$

5. $r=1,1,1,2,3$$\displaystyle r=1,1,1,2,3$

6. $r=2,\sqrt{2},-\sqrt{2}$$\displaystyle r=2,\sqrt{2},-\sqrt{2}$

7. $r=1+\sqrt{3},1-\sqrt{3}$$\displaystyle r=1+\sqrt{3},1-\sqrt{3}$

### Existence and Uniqueness

#### Theorem: Existence and Uniqueness Theorem

For any real numbers $a$$a$ ($\ne 0$$\ne 0$), $b$$b$, $c$$c$, ${t}_{0}$$t_0$, ${Y}_{0}$$Y_0$, and ${Y}_{1}$$Y_1$, there exists a unique solution to the IVP $a{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }+b{y}^{\phantom{\rule{0.167em}{0ex}}\prime }+cy=0$$\displaystyle ay^{\,\prime\prime}+by^{\,\prime}+cy=0$ on $\left(-\mathrm{\infty },\mathrm{\infty }\right)$$(-\infty,\infty)$, $y\left({t}_{0}\right)={Y}_{0}$$y(t_0)=Y_0$, ${y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left({t}_{0}\right)={Y}_{1}$$y^{\,\prime}(t_0)=Y_1$ and the solution is valid for all $t$$t$ in $\left(-\mathrm{\infty },\mathrm{\infty }\right)$$(-\infty,\infty)$.