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Math Help - Trying to get a general eqn, (a non linear 2nd order ODE) with 4 governing equations

  1. #1
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    Trying to get a general eqn, (a non linear 2nd order ODE) with 4 governing equations

    I have the following 4 governing equations

    i) V1(t) = a.Id(t)^2 + Rd.Id(t)
    ii) V1(t) = V(t) - R.I(t)
    iii) Ic(t)= C.dV(t)/dt
    iv) I(t) = Ic(t) + Id(t)

    The values of a, Rd, R and C are constant.

    And I need to find an equation in terms of V on the LHS and I on the RHS, but no matter how I look at it I can't find a solution. I've been two days looking at this and I must be missing something. everytime I try to solve it I start to get terms which are the product of I or dI/dt and V or dV/dt
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  2. #2
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    Re: Trying to get a general eqn, (a non linear 2nd order ODE) with 4 governing equat

    Quote Originally Posted by munkifisht View Post
    V1(t) = a.Id(t)^2 + Rd.Id(t)
    Kind of "confusing"...does a.Id(t)^2 mean a times Id (Id = one variable) times t^2 ?

    Can we let u = V1, v = Id and w = Rd, and rewrite that equation this way: ut = avt^2 + vwt ?
    Then it can be divided through by t, to become: u = avt + vw
    ????????????
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    Re: Trying to get a general eqn, (a non linear 2nd order ODE) with 4 governing equat

    Id(t)^2 simple means that Id which is a function of t is squared.
    might be clearer as [Id(t)]^2
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    Re: Trying to get a general eqn, (a non linear 2nd order ODE) with 4 governing equat

    I'm just going to rewrite this in a way I think is a little clearer, dump the constant terms (really not that important to this solution) and use the following terms
    X, Y, Z, J and K
    Where everything is a function of t

    I want to combine these equations into a single equation where the X, Y, and Z terms are no longer in the equation and there are only terms of J on the RHS and K on the LHS or visa versa

    i) X = Y^2 + Y
    ii) Z = dX/dt
    iii) J = Z + Y
    iv) K = X + J

    This should be possible but not matter which way I combine these I get something along the lines of

    K = J^2 + ( K'(t) + J'(t) )^2 - 2*J ( K'(t) + J'(t) ) - ( K'(t)+J'(t) )

    and obviously this is going to leave you with products of J and Ks that can't be seperated (I need all the K'(t) terms to be over with it's buddy on the LHS). No matter what stratagy I use to avoid this I can't help but get these terms combining.
    Last edited by munkifisht; May 30th 2012 at 05:21 AM.
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    Re: Trying to get a general eqn, (a non linear 2nd order ODE) with 4 governing equat

    Quote Originally Posted by munkifisht View Post
    i) X = Y^2 + Y
    ii) Z = dX/dt
    iii) J = Z + Y
    iv) K = X + J

    This should be possible but not matter which way I combine these I get something along the lines of

    K = J^2 + ( K'(t) + J'(t) )^2 - 2*J ( K'(t) + J'(t) ) - ( K'(t)+J'(t) )

    and obviously this is going to leave you with products of J and Ks that can't be seperated (I need all the K'(t) terms to be over with it's buddy on the LHS). No matter what stratagy I use to avoid this I can't help but get these terms combining.
    Quite a challenging thingy!
    Tried : k = K'(t) , j = J'(t)
    to get:
    K^2 - 4k^2 + 8k(J - j) = 4J^2 + 4j^2 - 8Jj - 4J - 1
    Close, but no ceegar...

    For curiosity, tried K = K'(t) , J = J'(t) ; so:
    K = J^2 + (K + J)^2 - 2J(K + J) - (K + J) ; leads to:
    J = K(K - 2)
    And that has endless integer solutions:
    J....K
    24,-4
    15,-3
    8,-2
    3,-1
    0,0
    -1,1
    0,2
    3,3
    8,4

    Hope that helps....probably not!
    Lots of others at this site more advanced than me...maybe one will step in
    Last edited by Wilmer; May 30th 2012 at 08:57 AM.
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  6. #6
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    Re: Trying to get a general eqn, (a non linear 2nd order ODE) with 4 governing equat

    Quote Originally Posted by Wilmer View Post
    K^2 - 4k^2 + 8k(J - j) = 4J^2 + 4j^2 - 8Jj - 4J - 1
    Made a goof; that should be:
    K + k(1 - k) + 2k(J - j) = J(J - 2j) + j(j - 1)
    Sorry...
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