hi all
how can i discuss the stability of the D.E
$\displaystyle
d^2x /dt^2 +alpha(alpha - 1) dx /dt + (pita - 1)(pita-2) x=0
$
thankx 4 help me
I know there is some cool way of doing stability problems.
I can just solve it.
$\displaystyle y''+\alpha(\alpha -1)y'+(\rho -1)(\rho -2)x=0$
Reduction of order,
$\displaystyle u=y'$
$\displaystyle u'+\alpha (\alpha -1)u+(\rho-1)(\rho-2)x=0$
$\displaystyle u'+\alpha (\alpha -1)u=-(\rho-1)(\rho-2)x$
The integrating factor is,
$\displaystyle \mu(x)=\exp \left( \int \alpha (\alpha -1) dx\right)=\exp(\alpha (\alpha -1)x)$
Thus, when we solve this differencial equation,
$\displaystyle u=e^{-\alpha(\alpha-1) x}\int ......... dx$
Thus, for stability we need that,
$\displaystyle \alpha(\alpha-1)>0$
$\displaystyle \alpha^2-\alpha >0$
$\displaystyle \alpha^2-\alpha+\frac{1}{4}>\frac{1}{4}$
$\displaystyle (\alpha-1/2)^2>1/4$
$\displaystyle \alpha-1/2>1/2 \mbox{ or }\alpha -1/2<-1/2$
$\displaystyle \alpha>1 \mbox{ or }\alpha <0$
BUT DO NOT RELY ON THIS. I just happend to know that the most important part in this is the exponential which is what I worked with.