Hi
Can anyone help me isolate x in the following equation:
x*cosh(L/x) = x+s
Maple outputs complex solutions, but I now its a real solution....
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Hi
Can anyone help me isolate x in the following equation:
x*cosh(L/x) = x+s
Maple outputs complex solutions, but I now its a real solution....
You can't isolate x, you need numerical methods to solve the equation.Quote:
Originally Posted by CyberCarsten
hi
Can you tell me how to do it numerical methods?
Hold on, I have to take some time to check my notes... I think I used to do this type of eqs. with a method other than Newton's. I'll be back to you a.s.a.p.Quote:
Originally Posted by CyberCarsten
What is the approximate range of values you're interested in? I should give actual values to L and s and then see what initial values I pick for checking the procedure I have in mind.
You can rewrite the equation in a modified fahsion:
Ch(L/x) = (x+s)/x
exp(L/x) + exp(-L/x) = 2(1 + s/x)
For convenience let y = L/x and multiply the whole equation by exp(y). Then:
exp(2y) -2(1 + sy/L)exp(y) + 1 = 0
Solving this 2nd degree equation where the unknown is exp(y) we arrive at these 2 solutions:
exp(y) = z + sqrt(z*z - 1)
and
exp(y) = z - sqrt(z*z + 1)
where z = 1 + sy/L
This means that if we plot exp(y) vs. z +/- sqrt(z*z - 1) the x corresponding to their crosspoint will be the sought value (see graph). In this graph I have plotted the left and right hand sides for s = L = 1. In this case it is obvious that there is a solution for the "positive" branch.
To determine it numerically you start by picking a value in the vicinity of the solution, say y0 = 1. Then you calculate z + sqrt(z*z - 1) (red curve) for z = 1 + sy0/L. Call this z1. Now derive y1 = (z1 - 1)L/s. Next find y2 such that y2 = exp(y1). For this y2 calculate z + sqrt(z*z - 1), and so on. You get ever closer to the solution.
I suppose it's a bit confusing. If you like I could explain the method using a much simpler equation.
Thank you so much! :)
It's been a pleasure! :)Quote:
Originally Posted by CyberCarsten
superb stuff sir!!!! bravo!!!