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.
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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.
Lottery is a tax on people who are bad at maths
If only mosquitoes sucked fat instead of blood...
To do is to be (Descartes). To be is to do (Sartre). To be do be do (Sinatra)
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.
Lottery is a tax on people who are bad at maths
If only mosquitoes sucked fat instead of blood...
To do is to be (Descartes). To be is to do (Sartre). To be do be do (Sinatra)
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.
Last edited by krtxmrtz; Jun 13th, 2005 at 11:44 AM.
Lottery is a tax on people who are bad at maths
If only mosquitoes sucked fat instead of blood...
To do is to be (Descartes). To be is to do (Sartre). To be do be do (Sinatra)
Lottery is a tax on people who are bad at maths
If only mosquitoes sucked fat instead of blood...
To do is to be (Descartes). To be is to do (Sartre). To be do be do (Sinatra)
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Originally Posted by CyberCarsten
