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Math Help - Exponential equation

  1. #1
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    Exponential equation

    I can't solve the equation (for variable E): e^{-\frac{E}{T}}*(T+E)=0.01aT.

    Help!


    Thanks!
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  2. #2
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    Can you explain the symbols in the equation?
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  3. #3
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    Sure, sorry!

    E is the unknown variable, a and T are known positive constant.
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  4. #4
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    Quote Originally Posted by Riesz View Post
    I can't solve the equation (for variable E): e^{-\frac{E}{T}}*(T+E)=0.01aT.

    Help!


    Thanks!
    That cannot be solved in terms of elementary functions. You can do this:

    Let u= T+ E. Then E= u- T so \frac{E}{T}= \frac{u}{T}- 1 and e^{-\frac{E}{T}}= e^{\frac{u}{T}e^{-T}

    The equation is now e^{-T}ue^{u/T}= 0.01aT or ue^{u/T}= 0.01aTe^T.

    Now, let v= u/T so that u= TV. The equation becomes Tve^v= 0.01aTe^T or ve^v= 0.01ae^T.

    Now you can use "Lambert's W function" which is defined as the inverse to the function f(x)= xe^x

    Then W(ve^v)= v= W(0.01ae^T).


    Then, of course, u= Tv= TW(0.01ae^T) and E= u- T= TW(0.01ae^T)- T or u= T(W(0.01ae^T)- 1).
    Last edited by mr fantastic; August 5th 2010 at 05:16 AM. Reason: Added a math tag.
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  5. #5
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    Let u=T+E.

    Then E=u-T, so \frac{E}{T}=\frac{u}{T}-1 and e^{-\frac{E}{T}}=e^{-\frac{u}{T}}*e^1

    The equation is now e^{-\frac{u}{T}}e^1*u=0.01aT.

    Now, let v=\frac{u}{T}, so that u=Tv.

    The equation becomes e^{-v}e*Tv=0.01aT, or e^{-v}v=0.01a*e^{-1}.


    Forgive me: where am I wrong?
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