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Thread: Need help with a basic equation

  1. #1
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    Need help with a basic equation

    Solve the equation $\displaystyle e^x - e^{-x} = a$ where a is an arbitrary real number.

    I have got a hint, set $\displaystyle e^x = t$, but unfortunately that does not help me much.
    Last edited by Sabo; Feb 10th 2010 at 06:26 AM.
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  2. #2
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    i think that it must be $\displaystyle e^x-e^{-x}=a$. Am i correct?
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  3. #3
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    Doh! Yes, you are right. I have edited the post.
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  4. #4
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    Quote Originally Posted by Sabo View Post
    Doh! Yes, you are right. I have edited the post.
    $\displaystyle e^x+e^{-x} = a \Leftrightarrow t+\frac{1}{t} = a \Leftrightarrow t^2-at+1=0$

    Now solve the quadratic equation
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  5. #5
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    Quote Originally Posted by Sabo View Post
    Doh! Yes, you are right. I have edited the post.
    Ahaa! That equation is well know. Now, i'll give you some help:

    $\displaystyle e^x+e^{-x}=a$

    Let be $\displaystyle e^x=t \implies e^{-x}= \dfrac{1}{t}$ then the equation transform to

    $\displaystyle t+\dfrac{1}{t}=a$

    Multiplying by t both sides:

    $\displaystyle t^2+1=at$

    can you finish it?
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  6. #6
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    Well, I guess it would go something along the lines of

    $\displaystyle t^2 -at + 1 = 0$
    $\displaystyle (t-\dfrac{a}{2})^2 - \dfrac{a^2+4}{4} = 0$
    $\displaystyle (t-\dfrac{a}{2})^2 = \dfrac{a^2+4}{4}$
    $\displaystyle t-\dfrac{a}{2} = +-\sqrt{\dfrac{a^2+4}{4}}$
    $\displaystyle t = \dfrac{a +- \sqrt{a^2+4}}{2}$

    For some reason I don't think that's the end of it though...
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  7. #7
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    now remember that $\displaystyle e^x>0$
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  8. #8
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    Quote Originally Posted by Sabo View Post
    Well, I guess it would go something along the lines of

    $\displaystyle t^2 -at + 1 = 0$
    $\displaystyle (t-\dfrac{a}{2})^2 - \dfrac{a^2+4}{4} = 0$
    $\displaystyle (t-\dfrac{a}{2})^2 = \dfrac{a^2+4}{4}$
    $\displaystyle t-\dfrac{a}{2} = +-\sqrt{\dfrac{a^2+4}{4}}$
    $\displaystyle t = \dfrac{a +- \sqrt{a^2+4}}{2}$

    For some reason I don't think that's the end of it though...
    Remember that you are serching for x, no t. Remember that, if you have $\displaystyle e^x=y \implies x=\ln(y)$
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  9. #9
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    Hmm...

    $\displaystyle t = e^x = \dfrac{a+-\sqrt{a^2+4}}{2} \implies x = \ln(\dfrac{a+-\sqrt{a^2+4}}{2})$

    I guess that can be considered a solution? Or? If it is, can we simplify it further?
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