Consider the second order homogeneous linear differential equation \begin{equation} x''+bx'+cx=0, \label{eq} \end{equation}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 (). Assuming a solution of this form, and substituting it into () gives$$ r^2 e^{rt}+bre^{rt}+ce^{rt}=0.$$ Thus \begin{equation} r^2+br+c=0 \label{char_eq} \end{equation}which is called the characteristic equation of (). Depending on the nature of the roots $r_1$ and $r_2$ of (), there are three cases.

• If the roots are real and distinct, then two linearly independent solutions of () are$$ x_1(t)=e^{r_1t},\quad x_2(t)=e^{r_2t}.$$
• If the roots are real and equal, then two linearly independent solutions of () are$$ x_1(t)=e^{r_1t},\quad x_2(t)=te^{r_1t}.$$
• If the roots are complex conjugates of the form $r_{1,2}=\alpha\pm i\beta$ , then two linearly independent solutions of () are$$ x_1(t)=e^{\alpha t}\cos \beta t,\quad x_2(t)=e^{\alpha t}\sin \beta t.$$

		(1)
		(2)

Also note that the only equilibrium of () is the origin $(0,0)$ . Suppose that $c\neq 0$ . Then $(0,0)$ is called a

1. source iff $b<0$ and $c>0$ ,
2. spiral source iff it is a source and $b^2-4c<0$ ,
3. sink iff $b>0$ and $c>0$ ,
4. spiral sink iff it is a sink and $b^2-4c<0$ ,
5. saddle iff $c<0$ ,
6. center iff $b=0$ and $c>0$ .