# second order linear differential equation with constant coefficients

 $x^{\prime\prime}+bx^{\prime}+cx=0,$ (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

 $r^{2}e^{rt}+bre^{rt}+ce^{rt}=0.$

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

 $x_{1}(t)=e^{r_{1}t},\quad x_{2}(t)=e^{r_{2}t}.$
• If the roots are real and equal, then two linearly independent solutions of (1) are

 $x_{1}(t)=e^{r_{1}t},\quad x_{2}(t)=te^{r_{1}t}.$
• If the roots are complex conjugates of the form $r_{1,2}=\alpha\pm i\beta$, then two linearly independent solutions of (1) are

 $x_{1}(t)=e^{\alpha t}\cos\beta t,\quad x_{2}(t)=e^{\alpha t}\sin\beta t.$

The general solution to (1) is then constructed from these linearly independent solutions, as

 $\phi(t)=C_{1}x_{1}(t)+C_{2}x_{2}(t).$ (3)

Characterizing the behavior of (3) can be accomplished by studying the two-dimensional linear system obtained from (1) by defining $y=x^{\prime}$:

 $\displaystyle x^{\prime}$ $\displaystyle=y$ (4) $\displaystyle y^{\prime}$ $\displaystyle=-by-cx.$ (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 $c\neq 0$. Then $(0,0)$ is called a

 $roman]{enumerate}\item\emph{source}iffb<0andc>0,\item\emph{spiral source}% iffitisasourceandb^{2}-4c<0,\item\emph{sink}iffb>0andc>0,\item\emph{spiral % sink}iffitisasinkandb^{2}-4c<0,\item\emph{}iffc<0,\item\emph{center}iffb=0andc% >0.\end{enumerate}\begin{flushright}\begin{tabular}[]{|ll|}\hline Title&second% order linear differential equation with constant coefficients\\ Canonical name&SecondOrderLinearDifferentialEquationWithConstantCoefficients\\ Date of creation&2013-03-22 13:24:49\\ Last modified on&2013-03-22 13:24:49\\ Owner&Mathprof (13753)\\ Last modified by&Mathprof (13753)\\ Numerical id&9\\ Author&Mathprof (13753)\\ Entry type&Topic\\ Classification&msc 34A30\\ Classification&msc 34-01\\ Classification&msc 34C05\\ Related topic&GeneralSolutionOfLinearDifferentialEquation\\ Related topic&TelegraphEquation\\ Defines&characteristic equation\\ Defines&source\\ Defines&sink\\ Defines¢er\\ \hline\end{tabular}\end{flushright}\end{document}$