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.
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.
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?
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...