second order linear differential equation with constant coefficients
Consider the second order homogeneous linear differential equation
$${x}^{\prime \prime}+b{x}^{\prime}+cx=0,$$  (1) 
where $b$ and $c$ are real constants.
The explicit solution is easily found using the characteristic equation^{} method. This method, introduced by Euler, consists in seeking solutions of the form $x(t)={e}^{rt}$ for (1). Assuming a solution of this form, and substituting it into (1) gives
$${r}^{2}{e}^{rt}+br{e}^{rt}+c{e}^{rt}=0.$$ 
Thus
$${r}^{2}+br+c=0$$  (2) 
which is called the characteristic equation of (1). Depending on the nature of the roots (http://planetmath.org/Equation) ${r}_{1}$ and ${r}_{2}$ of (2), there are three cases.

•
If the roots are real and distinct, then two linearly independent^{} solutions of (1) are
$${x}_{1}(t)={e}^{{r}_{1}t},{x}_{2}(t)={e}^{{r}_{2}t}.$$ 
•
If the roots are real and equal, then two linearly independent solutions of (1) are
$${x}_{1}(t)={e}^{{r}_{1}t},{x}_{2}(t)=t{e}^{{r}_{1}t}.$$ 
•
If the roots are complex conjugates^{} of the form ${r}_{1,2}=\alpha \pm i\beta $, then two linearly independent solutions of (1) are
$${x}_{1}(t)={e}^{\alpha t}\mathrm{cos}\beta t,{x}_{2}(t)={e}^{\alpha t}\mathrm{sin}\beta t.$$
The general solution to (1) is then constructed from these linearly independent solutions, as
$$\varphi (t)={C}_{1}{x}_{1}(t)+{C}_{2}{x}_{2}(t).$$  (3) 
Characterizing the behavior of (3) can be accomplished by studying the twodimensional linear system obtained from (1) by defining $y={x}^{\prime}$:
${x}^{\prime}$  $=y$  (4)  
${y}^{\prime}$  $=bycx.$  (5) 
Remark that the roots of (2) are the eigenvalues^{} of the Jacobian matrix of (5). This generalizes to the characteristic equation of a differential equation^{} of order $n$ and the $n$dimensional system associated to it.
Also note that the only equilibrium of (5) is the origin $(0,0)$. Suppose that $c\ne 0$. Then $(0,0)$ is called a
$$ 