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

Definition: General Solution

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

${y}_{g}(t)={c}_{1}{y}_{1}(t)+{c}_{2}{y}_{2}(t)$

Investigation 03

Let ${y}_{1}(t)$ and ${y}_{2}(t)$ be linearly-independent solutions to $a{y}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime \prime}(t)+b{y}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(t)+cy(t)=0$. Show that ${c}_{1}{y}_{1}(t)+{c}_{2}{y}_{2}(t)$ is also a solution to $a{y}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime \prime}(t)+b{y}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(t)+cy(t)=0$.

Solutions to Linear Homogeneous ODEs

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

If ${r}_{1}$ and ${r}_{2}$ are distinct real roots of the characteristic equation, then the general solution is ${y}_{g}(t)={c}_{1}{e}^{{r}_{1}t}+{c}_{2}{e}^{{r}_{2}t}$;

${r}_{1}$ is a repeated root of the characteristic equation, then the general solution is ${y}_{g}(t)={c}_{1}{e}^{{r}_{1}t}+{c}_{2}t{e}^{{r}_{1}t}$;

${r}_{1}$ and ${r}_{2}$ 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, ${y}_{h}(x)$.

$r=-2,1$

$r=0,2,3$

$r=-1,-1,1$

$r=-3,0,0,2$

$r=1,1,1,2,3$

$r=2,\sqrt{2},-\sqrt{2}$

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

Existence and Uniqueness

Theorem: Existence and Uniqueness Theorem

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