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Math Help - Formulating a second-order ordinary differential equation as a first-order system?

  1. #1
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    Formulating a second-order ordinary differential equation as a first-order system?

    Hi, i need some help on this second-order ODE equation:
    mu'' + f(u') + s(u) = F(t), t > 0, u(0) = U0, u'(0) = V0
    How would it look like as a first-order system?
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  2. #2
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    Quote Originally Posted by !alex! View Post
    Hi, i need some help on this second-order ODE equation:
    mu'' + f(u') + s(u) = F(t), t > 0, u(0) = U0, u'(0) = V0
    How would it look like as a first-order system?

    hey mate,

    there are numerous ways to achieve this, but would probably be the simplest way is to let

    w(t) = u'(t) --> w'(t) = u''(t)
    Thus,
    mu''(t)+f(u'(t)) + s(u(t)) = F(t) becomes,

    mw'(t) + f(w(t)) + u(t) = F(t) , u(0) = U0 u'(0) = w(0) = V0

    here we observe, we have reduced the order of the differential equation, and our system becomes,

    mw'(t) + f(w(t)) + u(t) = F(t)
    w(t) - u'(t) = 0

    Subject to, u(0) = U0 , w(0) = V0

    Hope this helps,

    let me know if you require any further assistance,

    Regards,

    David
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  3. #3
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    Thanks mate, I really appreciate it. I'm going to look better at it later and try to understand it. Right now i just have to use it in some other exercise.
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