Using Hyperbolic functions, Solve sinh(2x) = 8cosh(x). Give answer in ln.

[Bobby Bones]

I have used 2sinhx.coshx = sinh(2x)

2sinhx.coshx = 8coshx

Then divided by coshx:

2sinhx = 8

Divided by 2:

sinhx = 4

Substituted sinhx = (e^x - e^-x)/2:

So now I have: (e^x - e^-x) = 8

How do I convert this to ln/solve from here. Thanks.

 

[Deleted member 4993]

I have used 2sinhx.coshx = sinh(2x)

2sinhx.coshx = 8coshx

Then divided by coshx:

2sinhx = 8

Divided by 2:

sinhx = 4

Substituted sinhx = (e^x - e^-x)/2:

So now I have: (e^x - e^-x) = 8

How do I convert this to ln/solve from here. Thanks.

(e^x - e^-x) = 8

(e^x)^2 - (8*e^x) - 1 = 0 ...................... continue......

 

[Bobby Bones]

(e^x - e^-x) = 8

(e^x)^2 - (8*e^x) - 1 = 0 ...................... continue......

Could you please explain how to reached this step and how do I convert it to ln?

I failed at factorizing it like a normal quadratic eqtn.

 

[pka]

I have used 2sinhx.coshx = sinh(2x)
So now I have: (e^x - e^-x) = 8

From there
\(\displaystyle \begin{align*}(e^x - e^{-x}) &=8 \\e^{2x}-8e^x-1&=0 \end{align*}\)
That is a quadratic equation in \(\displaystyle e^x\) so that
\(\displaystyle e^x=\dfrac{8\pm\sqrt{64+4}}{2}\)
Be reminded that \(\displaystyle e^x>0\).

 

[Bobby Bones]

Thanks to both of you for your help.

My final answer is x=ln(4x + sqrt(17))

 

[Deleted member 4993]

Thanks to both of you for your help.

My final answer is x=ln(4x + sqrt(17))

Almost correct!

What is that "x" doing there?

 

[Bobby Bones]

Almost correct!

What is that "x" doing there?

Would you believe it? I've been looking for x all day and when I finally find it, his friend turns up too ?

Thanks again.