Solving Hyperbolic Equations Using Exponential Definitions

[2^Oscar]

Hey guys,

I was doing some work on hyperbolic functions and teaching myself to solve some equations. One of the questions in the book really has me stumped:

Express using exponential definitions of cosh(x) and sinh(x) find the exact solution of:
tanh(x) + sinh(x) = 3

I had a go at solving it and this is how far I got:

2tanh(x) + 2sinh(x) = 6

2(e^2x -1)/(e^2x -1) + e^x - e^-x = 6

e^3x - 4e^2x - 8 - e^-x = 0

e^4x - 4e^3x - 8e^x - 1 = 0


then if y=e^x

y⁴ - 4y³ - 8y -1 = 0


After this I get stuck. I can't find any factors in order to solve it using factor theorem so I'm guessing I'm going to get some weird solutions - but the question asks specifically for exact answers?

Would anyone mind please helping me out? (I really hope I haven't made some pathetic little mistake but I really can't see anything...)


Thanks in advance,
Oscar

 

[uart]

Your working looks correct and a numerical solution would be easy enough but yeah I can't see any way to get an "exact" solution here either.

Here's another approach which you may find useful (btw it leads to a slightly simpler equation but still no "exact" solution that I can see - though maybe someone else will).

Just about every trig identity for standard trig functions has a counterpart for the hyperbolic trig's. In this case it's the hyperbolic counterpart of the t=tan(x/2) identities that are useful. These identies are :

t = \tanh(x/2)

\tanh(x) = 2t/(1+t^2)

\sinh(x) = 2t/(1-t^2)

\cosh(t) = (1+t^2)/(1-t^2)

Using the identities for tanh and sinh above in your equation it's pretty easy to obtain :

3t^4 + 4t - 3 = 0

Numerically t = 0.63106 and x = 2 \tanh^{-1}(t) = 1.4863 to 5 sig figures. This solution also works in your equation (and of course in the original hyperbolic equation) so we can safely assume it is "correct" - though unfortunately just a numerical solution.