# Need help with a basic equation

• Feb 10th 2010, 06:10 AM
Sabo
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.
• Feb 10th 2010, 06:18 AM
felper
i think that it must be $\displaystyle e^x-e^{-x}=a$. Am i correct?
• Feb 10th 2010, 06:26 AM
Sabo
Doh! Yes, you are right. I have edited the post.
• Feb 10th 2010, 06:28 AM
danielomalmsteen
Quote:

Originally Posted by Sabo
Doh! Yes, you are right. I have edited the post.

$\displaystyle e^x+e^{-x} = a \Leftrightarrow t+\frac{1}{t} = a \Leftrightarrow t^2-at+1=0$

• Feb 10th 2010, 06:28 AM
felper
Quote:

Originally Posted by Sabo
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?
• Feb 10th 2010, 06:48 AM
Sabo
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...
• Feb 10th 2010, 06:52 AM
danielomalmsteen
now remember that $\displaystyle e^x>0$
• Feb 10th 2010, 06:53 AM
felper
Quote:

Originally Posted by Sabo
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 $\displaystyle e^x=y \implies x=\ln(y)$
• Feb 10th 2010, 07:21 AM
Sabo
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?