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

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- Jan 8th 2007, 12:33 PMsweetstability of the differential equation
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 - Jan 8th 2007, 01:21 PMThePerfectHacker
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. - Jan 9th 2007, 12:56 AMsweet
thank u ThePerfectHacker

but i need to study the stability in each piont of the ]0,1[