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Math Help - How to convert this 2nd order ODE to a 1st order standard form

  1. #1
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    How to convert this 2nd order ODE to a 1st order standard form

    Convert to standard form ODE system y' = f(t, y):
    (t^2)y'' + sin(y') + 2y − 1 = 0

    the goal is to reduce the equation above to be a first order ode.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by oxxiissiixxo View Post
    Convert to standard form ODE system y' = f(t, y):
    (t^2)y'' + sin(y') + 2y − 1 = 0

    the goal is to reduce the equation above to be a first order ode.
    Hmm...

    I know that if you make the substitutions u_1=y and u_2=\dot{y}, you will see that \dot{u_1}=\dot{y}=u_2 and \dot{u_2}=\ddot{y}

    Thus, you get the first order system

    \left\{\begin{array}{lcl}<br />
\dot{u_1}=u_2\\ \dot{u_2}=t^{-2}-2u_1t^{-2}-t^{-2}\sin\left(u_2\right)\end{array}\right.

    Does this make sense?
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  3. #3
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    thz
    Last edited by oxxiissiixxo; January 10th 2009 at 06:49 PM.
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