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Solving nonlinear equation with integrals and beta functions

Muhammad Fahem Bin Musa
Posted 1 year ago

enter image description here

Hello! I am new to Mathematica. Does anybody here know how to solve (for alpha) this equation? I find it very difficult. I have tried to use NSolve[ ] and Solve[ ]. But it does not work.
POSTED BY: Muhammad Fahem Bin Musa
6 Replies
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Here is a way:

Clear[alpha, f];
f[alpha_] =
 (1 - 4  Integrate[t^(1 - 1/alpha) (1 - t)^(1/alpha),
      {t, 0, 1/2}, Assumptions -> alpha > 1]/
     Integrate[t^(-1/alpha) (1 - t)^(1/alpha),
      {t, 0, 1/2}, Assumptions -> alpha > 1])
Plot[f[alpha], {alpha, 1, 4}, GridLines -> {None, {.2455}}]
Solve[f[alpha] == .2455 && 2 < alpha < 3, alpha]
POSTED BY: Gianluca Gorni
Bill Nelson
Bill Nelson
Posted 1 year ago
POSTED BY: Bill Nelson

Thank you! You have helped me a lot. BTW, what do you mean by " there are always at least six different ways of doing anything in Mathematica." ?
POSTED BY: Muhammad Fahem Bin Musa
Bill Nelson
Bill Nelson
Posted 1 year ago

New users may not understand why the method they tried to use to solve a problem did not work. And they are sometimes surprised when they are given several completely different ways of solving the problem. Sometimes there may be as many as ten completely different ways of solving a problem. Some of those may be very difficult to understand.
POSTED BY: Bill Nelson
POSTED BY: Muhammad Fahem Bin Musa
Bill Nelson
Bill Nelson
Posted 1 year ago

Try

Assuming[alpha>1,expr=1-2455/10^4==4Integrate[t^(1-1/alpha)(1-t)^(1/alpha),{t,0,1/2}]Integrate[t^-(1/alpha)(1-t)^(1/alpha),{t,0,1/2}]]
FindRoot[expr,{alpha,2}]

and

FindInstance[1-2455/10^4==4Integrate[t^(1-1/alpha)(1-t)^(1/alpha),{t,0,1/2}]Integrate[t^-(1/alpha)(1-t)^(1/alpha),{t,0,1/2}]&&alpha>2,alpha]

There should be ways to get NSolve and Solve to do this for you.

And there will be even more ways of doing this.
POSTED BY: Bill Nelson
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