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

${\displaystyle a{y}^{\prime \prime }+b{y}^{\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

${\displaystyle a_n{y}^{(n)}+a_{n-1}{y}^{(n-1)}+\cdots +a_2{y}^{\prime \prime }+a_1{y}^{\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 ${\displaystyle 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 ${\displaystyle a{y}^{\prime \prime }+b{y}^{\prime }+cy=0}$$\displaystyle ay^{\,\prime\prime}+by^{\,\prime}+cy=0$, $a\ne 0$$a\ne 0$ on $(-\mathrm{\infty },\mathrm{\infty })$$(-\infty,\infty)$, then the general solution to the homogeneous linear DE is a linear combination of $y_1$$y_1$ and $y_2$$y_2$:

$y_g(t)=c_1{y}_1(t)+c_2{y}_2(t)$$y_g(t)=c_1y_1(t)+c_2y_2(t)$

Investigation 03

Let $y_1(t)$$y_1(t)$ and $y_2(t)$$y_2(t)$ be linearly-independent solutions to $a{y}^{\prime \prime }(t)+b{y}^{\prime }(t)+cy(t)=0$$ay^{\,\prime\prime}(t)+by^{\,\prime}(t)+cy(t)=0$. Show that $c_1{y}_1(t)+c_2{y}_2(t)$$c_1y_1(t)+c_2y_2(t)$ is also a solution to $a{y}^{\prime \prime }(t)+b{y}^{\prime }(t)+cy(t)=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(t)=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$$r_1$ is a repeated root of the characteristic equation, then the general solution is $y_g(t)=c_1{e}^{r_{1}t}+c_{2}t{e}^{r_{1}t}$$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(x)$$y_h(x)$.

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

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

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

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

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

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

7. ${\displaystyle 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$$Y_1$, there exists a unique solution to the IVP ${\displaystyle a{y}^{\prime \prime }+b{y}^{\prime }+cy=0}$$\displaystyle ay^{\,\prime\prime}+by^{\,\prime}+cy=0$ on $(-\mathrm{\infty },\mathrm{\infty })$$(-\infty,\infty)$, $y(t_0)=Y_0$$y(t_0)=Y_0$, $y^{\prime }(t_0)=Y_1$$y^{\,\prime}(t_0)=Y_1$ and the solution is valid for all $t$$t$ in $(-\mathrm{\infty },\mathrm{\infty })$$(-\infty,\infty)$.

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Last Modified: Monday, 6 September 2020 17:33 EDT