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Math Help - Formulating a 2nd-order ODE as a system.

  1. #1
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    Formulating a 2nd-order ODE as a system.

    I am working on an assignment, and my first task is to write this second order differential equation as a system of two first order equations. I have done this previously for special cases where f and u are linear functions (typically f(u') = b*u' and s(u) = ku), but I have no idea of what to do with this generalized equation. Can somebody help me?

    mu ̈+f(u ̇)+s(u)=F(t), t>0, u(0)=U0, u ̇(0)=V0
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  2. #2
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    Can you please rewrite your equation? I can't read some of the symbols. They come out as gobbledy-gook.
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  3. #3
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    mu'' + f(u') + s(u) = F(t) , t > 0, u(0) = U0, u'(0) = V0
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  4. #4
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    The same basic procedure works. You let y_{1}=u, y_{2}=u'.

    Then, you can say that y_{1}'=y_{2}, and y_{2}'=?

    If you fill in the question mark there from the original DE, you'll be done with this step.
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  5. #5
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    is y'2 = (F(t) - f(u') - s(u))/m then?
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  6. #6
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    Well, it's really

    y_{2}'=\dfrac{F(t)-f(u')-s(u)}{m}, with the prime on y_{2},

    but then you can substitute in for u and u', right? What do you get? (No u's allowed in final answer!)
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  7. #7
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    Since u = y1 and u' = y2, I get: (F(t) - f(y2) - s(y1)) / m
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  8. #8
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    So writing out your system in full gives you what?
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