Sinh 2x=8 coshx

sinh2x =8coshx

2sinhx coshx - 8coshx = 0
2coshx(sinhx-4) = 0
coshx=0 (no solutions)
sinhx = 4
x = log(4+√17)

http://www.wolframalpha.com/input/?i=sinh2x%3D8coshx

To solve the equation sinh(2x) = 8cosh(x), we can start by rewriting the hyperbolic functions in terms of exponentials.

The hyperbolic functions sinh(x) and cosh(x) can be defined using exponentials as follows:

sinh(x) = (e^x - e^(-x)) / 2
cosh(x) = (e^x + e^(-x)) / 2

Using these definitions, we can rewrite the given equation as:

(e^(2x) - e^(-2x))/2 = 8 * (e^x + e^(-x))/2

Next, we can multiply both sides of the equation by 2 to eliminate the denominators:

e^(2x) - e^(-2x) = 16 * (e^x + e^(-x))

Now, let's simplify the equation further. Let's define a new variable, y, as e^x:

y^2 - 1/y^2 = 16 * (y + 1/y)

Multiplying both sides by y^2, we obtain:

y^4 - 1 = 16 * (y^3 + y)

Rearranging the equation, we have:

y^4 - 16y^3 + y - 16 = 0

Now, we can solve this quartic equation for y. There isn't a simple algebraic method for solving quartic equations, but you can find the roots using numerical methods or software.

Once you have obtained the values of y, you can substitute them back into the equation y = e^x and solve for x. Remember that e^x can have both positive and negative values, so you may get multiple solutions for x.