I have an equation in two variables and I am trying to find if there exists some range where the equation is satisfied. Having tried a few methods I am not yet sure if I can say that no such range exists or simply that I can't find it.

Here is the equation :

Binomial[n,2]*(3.97887*10^-10 Cos[200 k ArcSinh[50/k]])/Sqrt[1 + 2500/k^2] = -0.00003

Kindly provide some suggestions.

Edit :

If it is somehow helpful then I am looking for extremely high values of n (could be 10^23 order) which can satisfy this equation with some range of k.


1 Answer 1


For such great n try to rescale your equation!

Try n->Exp[logn],k->Exp[-logk] with scaled variables logk,logn:

eq = (Binomial[n, 2]*(3.97887*10^-10 Cos[200 k ArcSinh[50/k]])/Sqrt[1 + 2500/k^2] == -0.00003 // Rationalize) /. {n -> Exp[logn], k -> Exp[-logk]}

soln[logk_] :=logn /. NSolve[{(1.9894350000000002`*^-10 E^logn (-1 + E^logn) Cos[200 E^-logk ArcSinh[50 E^logk]])/Sqrt[1 + 2500 E^(2 logk)] == -(3/100000), -25 < logn < 25}, logn, Reals] [[1]]

This gives

Plot[soln[logk], {logk, 0, .1},AxesLabel-> {"Log[k]","Log[n]"}]

enter image description here

  • $\begingroup$ Just a little more clarification needed. ReplaceAll gives an error : ReplaceAll::reps: {(1.98944*10^-10 E^logn (-1+E^logn) Cos[200 E^-logk ArcSinh[50 Power[<<2>>]]])/Sqrt[1+2500 E^Times[<<2>>]]==-(3/100000),-25<logn<25} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing. Given this error, is the plot okay? Also, would you say that n and k don't really have any solutions? Given the almost tangent curves or zeroes? $\endgroup$
    – Nitin
    Apr 24, 2020 at 20:12
  • $\begingroup$ Where does this error come from? I retested my code with fresh kernel, seems to be ok. $\endgroup$ Apr 24, 2020 at 20:20
  • $\begingroup$ It's showing up on my Mathematica 11.0 $\endgroup$
    – Nitin
    Apr 24, 2020 at 20:23
  • $\begingroup$ It occurs when you try to plot? Does a single evaluation soln[.1] (*7.97209*) evaluate? My version is Mathematica 12. $\endgroup$ Apr 24, 2020 at 20:25
  • $\begingroup$ Now, it's 7.97209 without any error $\endgroup$
    – Nitin
    Apr 24, 2020 at 20:34

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