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

  $$x_1(t)=e^{\alpha t}\cos \beta t,x_2(t)=e^{\alpha t}\sin \beta t.$$

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 two-dimensional linear system obtained from (1) by defining $y=x^{\prime }$:

$x^{\prime }$ $=y$ (4)

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