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Thread: stability of the differential equation

  1. #1
    Junior Member sweet's Avatar
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    stability 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
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  2. #2
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    Quote Originally Posted by sweet View Post
    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.
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  3. #3
    Junior Member sweet's Avatar
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    thank u ThePerfectHacker

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