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Homesecond order linear differential equation with constant coefficients
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second order linear differential equation with constant coefficients
Consider the second order homogeneous linear differential equation
$x^{{\prime\prime}}+bx^{{\prime}}+cx=0,$  (1) 
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 $r_{1}$ 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 twodimensional linear system obtained from (1) by defining $y=x^{{\prime}}$:
$\displaystyle x^{{\prime}}$  $\displaystyle=y$  (4)  
$\displaystyle y^{{\prime}}$  $\displaystyle=bycx.$  (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.
Mathematics Subject Classification
34A30 no label found3401 no label found34C05 no label found Forums
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