Hyperbolic tangents ate my variable! [SOLVED]

[LordMantir]

I am in the unfortunate situation of having to plot a graph of the following form:

y = k_1 tanh(\frac{k_2 \cdot x}{y})

Is it possible to get the y outside the tanh without swallowing the other y up in an inverse hyperbolic tangent? My maths (starting Upper VIth double maths) doesn't stretch this far.

EDIT: Solved by plotting x = f(y). Not pretty but it works.

[rhino]

EDIT:

Do you have to find y as a function of x in order to be able to plot?

(is this homework?)

[Macbi]

You can certainly find x as a function of y, does that help?

[LordMantir]

This is a possible extension to my A2 physics coursework. At the moment I am using the high- and low-value approximations for tanh x to plot two graphs on the same axes. So, kind of homework. The context is wave speed formulae: the actual formula I would like to plot is:

c = \frac{g}{2f\pi}tanh(\frac{2df\pi}{c})

for constant f, g and varying d. I could certainly, given enough time, plot an approximation using an iterative process, using last iteration's y values each iteration, but that's fiddly and also I don't have software available that will do that nicely. Getting x as a function of y is a possibility but not ideal.

If it's not easily done then the approximations are fine, but being a mathematician at heart I wanted to know if it was possible, hopefully piggybacking on someone else's experience of similar situations without trying to do it and ending up down a dead end and using up time that could be more productively spent actually finishing the coursework. After the deadline I will most likely have a better go at this myself - this is an interim solution, asking you guys for help!

[rhino]

If you're hoping to rearrange the equation into something of the form y = f(x) then I can show that it can't be done:

Note that tanh is an odd function, in that tanh(-x) = -tanh(x).
From this it very quickly follows that if (x₀,y₀) solves your equation, then so does (x₀,-y₀). That means that y is not a function of x.
However, this is probably a "silly" problem because I assume that from the physical context, only one solution will make sense.

More worrying (though I can't prove it) is the possibility that you might not be able to obtain something like "y = plus or minus f(x)" where f is a "friendly" function built out of exponentials, trigonometric functions, polynomials and logs.

The good news is that you don't need to do this if all you're trying to do is draw an accurate graph (re-read the two posts above!)

[BlackSails]

You could expand tanh in a taylor series, and take a bunch of terms.

[Averazul]

You can rearrange for x = f(y), graph that, then flip the graph along the line y = x