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Math Help - How to simplify an equation?

  1. #1
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    How to simplify an equation?

    Hello,
    I have the following problem. There are known dependencies between variables.


    a = a1 + a2.

    C = C1 + C2.

    C = \frac{a}{{{{\left( {1 + d} \right)}^t}}},C1 = \frac{{a1}}{{{{\left( {1 + d1} \right)}^t}}},C2 = \frac{{a2}}{{{{\left( {1 + d2} \right)}^t}}}.

    The desired quantity is d. In this simple case it can be easily expressed using C1, C2, C, d1, d2. The solution is following

    {\left( {1 + d} \right)^t} = \frac{a}{C} = \frac{{a1 + a2}}{C} = \frac{{{{\left( {1 + d1} \right)}^t}C1 + {{\left( {1 + d2} \right)}^t}C2}}{C}.

    d = {\left( {\frac{{{{\left( {1 + d1} \right)}^t}C1 + {{\left( {1 + d2} \right)}^t}C2}}{C}} \right)^{\frac{1}{t}}} - 1.

    But I am confused with the similar, but a bit more complex problem.
    The known dependencies between variables are following

    {a_t} = a{1_t} + a{2_t}.

    {C_t} = C{1_t} + C{2_t}.

    {C_{t - 1}} = \frac{{{a_t}}}{{{{\left( {1 + {d_t}} \right)}^t}}} + \frac{{{C_t}}}{{\left( {1 + {d_t}} \right)}},C1 = \frac{{a{1_t}}}{{{{\left( {1 + d{1_t}} \right)}^t}}} + \frac{{C{1_t}}}{{\left( {1 + d{1_t}} \right)}},C2 = \frac{{a2{}_t}}{{{{\left( {1 + d{2_t}} \right)}^t}}} + \frac{{C{2_t}}}{{\left( {1 + d{2_t}} \right)}}.

    t - time index.

    The desired quantity is {d_t} .

    The goal is to express {d_t}

    using C1, C2, C, d1, d2, a1 and a2. The solution of the equation seems cumbersome. Is it possible to reduce the resulting formula to elegant form, which would resemble d = {\left( {\frac{{{{\left( {1 + d1} \right)}^t}C1 + {{\left( {1 + d2} \right)}^t}C2}}{C}} \right)^{\frac{1}{t}}} - 1?
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  2. #2
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    Re: How to simplify an equation?

    Hey tabibito.

    Does t have a range (i.e. is it say greater than zero)?

    If it does, then expand everything out in terms of C1_0 and C2_0 and given these expressions see if you can use a simplification much like the one you used above.

    Also does d_t depend on the time t or is constant across all values of time?
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  3. #3
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    Re: How to simplify an equation?

    Thank you for the reply, Chiro. t>0. It can be a fraction, say, 1/12 or 1/2 etc.

    d_t depends on C1_t-1, C2_t-1, C_t-1, d1_t, d2_t, a1_t and a2_t. As these variables are time-dependent, d_t will also change with time.

    I tried replacements, but it seems that the equation resists simplification.

    {C_{t - 1}} = \frac{{{a_t} + {C_t}{{\left( {1 + {d_t}} \right)}^{t - 1}}}}{{{{\left( {1 + {d_t}} \right)}^t}}}.

    {\left( {1 + {d_t}} \right)^t} = \frac{{{a_t} + {C_t}{{\left( {1 + {d_t}} \right)}^{t - 1}}}}{{{C_{t - 1}}}}.

    a{1_t} = C{1_{t - 1}}{\left( {1 + d{1_t}} \right)^t} - \frac{{C{1_t}{{\left( {1 + d{1_t}} \right)}^t}}}{{\left( {1 + d{1_t}} \right)}}.

    {\left( {1 + {d_t}} \right)^t} = \frac{{C{1_{t - 1}}{{\left( {1 + d{1_t}} \right)}^t} - \frac{{C{1_t}{{\left( {1 + d{1_t}} \right)}^t}}}{{\left( {1 + d{1_t}} \right)}} + C{2_{t - 1}}{{\left( {1 + d{2_t}} \right)}^t} - \frac{{C{2_t}{{\left( {1 + d{2_t}} \right)}^t}}}{{\left( {1 + d{2_t}} \right)}} + {C_t}{{\left( {1 + {d_t}} \right)}^{t - 1}}}}{{{C_{t - 1}}}}.
    Last edited by tabibito; August 27th 2013 at 09:36 PM.
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  4. #4
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    Re: How to simplify an equation?

    Sorry. I mistyped in the first post

    It should be

    {C_{t - 1}} = \frac{{{a_t}}}{{{{\left( {1 + {d_t}} \right)}^t}}} + \frac{{{C_t}}}{{\left( {1 + {d_t}} \right)}},C{1_{t - 1}} = \frac{{a{1_t}}}{{{{\left( {1 + d{1_t}} \right)}^t}}} + \frac{{C{1_t}}}{{\left( {1 + d{1_t}} \right)}},C{2_{t - 1}} = \frac{{a2{}_t}}{{{{\left( {1 + d{2_t}} \right)}^t}}} + \frac{{C{2_t}}}{{\left( {1 + d{2_t}} \right)}}.
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