Mathematica won't solve: $(H-T)^x-(L+T)^x=x*c/p$

Question

I'm trying to solve the equation:

((H-T)^x-(L+T)^x)/x==c/p

for T. I've tried working with Solve and Reduce and I've also tried restricting the solution to Real values only but nothing seems to work. Also, I'm not sure how to use the fact that:

Reduce[((H - T)^x - (L + T)^x)/x == c/p && p>= 0 && p <= 1 && x > 0 && x <= 1, T]

I also tried:

Solve[((H - T)^x - (L + T)^x)/x == c/p && p >= 0 && p <= 1 && x > 0 && x <= 1, T] 

Any suggestions?

Answer 1

When symbolic methods are not up to the task, one can try numerical methods, either NSolve or FindRoot.

One can make a function in a number of ways. For example, using NSolve:

solT[H_?NumericQ, L_?NumericQ, x_?NumericQ, c_?NumericQ, p_?NumericQ] /;
       p >= 0 && p <= 1 && x > 0 && x <= 1 :=              (* restrict domain *)
 NSolve[((H - T)^x - (L + T)^x)/x == c/p, T, Reals]

We can get T as a function of the parameters with

First[T /. solT[..]]

One problem is that, in general, there may not be exactly one solution. One could use the following instead of First. It will give a warning, and one could adapt the behavior as desired.

firstOfOne::many = "There are `` parts; expected 1";
firstOfOne[e_] := (If[Length[e] != 1,
    Message[firstOfOne::many, Length[e]]
    ];
   First[e]);

Examples:

solT[10, 2, 0.4, 3, 1/2]
(*  {{T -> -1.9517}}  *)

Plot[firstOfOne[T /. solT[10, 2, x, 3, 1/2]], {x, 0, 1}]

There does not seem to be a case where there is more than one solution returned in my somewhat random testing, so maybe firstOfOne might be overkill.