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second order linear differential equation with constant coefficients (Topic)

Consider the second order homogeneous linear differential equation

$\displaystyle x''+bx'+cx=0,$ (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

$\displaystyle r^2 e^{rt}+bre^{rt}+ce^{rt}=0. $
Thus
$\displaystyle r^2+br+c=0$ (2)

which is called the characteristic equation of (1). Depending on the nature of the roots $ 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
    $\displaystyle 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 (1) are
    $\displaystyle 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 (1) are
    $\displaystyle 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
$\displaystyle \phi(t)=C_1x_1(t)+C_2x_2(t).$ (3)

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

$\displaystyle x'$ $\displaystyle =y$ (4)
$\displaystyle y'$ $\displaystyle =-by-cx.$ (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

  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$.



"second order linear differential equation with constant coefficients" is owned by Mathprof. [ full author list (3) | owner history (2) ]
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See Also: general solution of linear differential equation, telegraph equation

Also defines:  characteristic equation, source, sink, center

Attachments:
harmonic oscillator (Application) by perucho
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Cross-references: iff, origin, order, differential equation, Jacobian matrix, eigenvalues, linear system, general solution, complex conjugates, linearly independent, roots, Euler, solution, real, linear differential equation, homogeneous, second order
There are 11 references to this entry.

This is version 6 of second order linear differential equation with constant coefficients, born on 2003-02-02, modified 2007-01-28.
Object id is 3960, canonical name is SecondOrderLinearDifferentialEquation.
Accessed 17703 times total.

Classification:
AMS MSC: 34-01 (Ordinary differential equations :: Instructional exposition )
 34A30 (Ordinary differential equations :: General theory :: Linear equations and systems, general)
 34C05 (Ordinary differential equations :: Qualitative theory :: Location of integral curves, singular points, limit cycles)
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