Can I solve a certain equation?
Manipulate[
Show[
ContourPlot[
(1 + x)^(2/m) - (1 - x)^(2/m) - (1 - x^2)^(1/m),
{x, -.1, 1.1}, {m, .1, 5}, Contours -> {0}],
pt = {x /. FindRoot[
(1 + x)^(2/m) - (1 - x)^(2/m) - (1 - x^2)^(1/m),
{x, m/5}],
m};
Graphics[{
Text[ToString[Round[pt, .001]], pt, {-1.25, 0}],
Red, AbsolutePointSize[6], Point[pt]}]],
{{m, 1}, .1, 5, .01, Appearance -> "Labeled"}]
This problem is solvable analytically, although we use Mathematica for some of the algebra. To begin, write the initial expression as
(1 + x)^(2/m) - (1 - x)^(2/m) - (1 - x^2)^(1/m);
and divide through by the third term.
Expand[-%/%[[3]]] /. (1 - x^2)^(-1/m) -> (1 - x)^(-1/m) (1 + x)^(-1/m)
(* -1 - (1 - x)^(1/m) (1 + x)^(-1/m) + (1 - x)^(-1/m) (1 + x)^(1/m) *)
Replacing ((1 + x)/(1 - x))^(1/m)
by z
casts the expression into the form
z - 1 - 1/z
which can be solved by
zsol = Solve[z - 1 - 1/z == 0, z]
(* {{z -> 1/2 (1 - Sqrt[5])}, {z -> 1/2 (1 + Sqrt[5])}} *)
Choose the second solution to obtain real x
.
zsol[[2, 1]] /. z -> ((1 + x)/(1 - x))^(1/m) /. Rule -> Equal;
Solve[%, x][[1, 1]]
(* x -> (-1 + (1/2 + Sqrt[5]/2)^m)/(1 + (1/2 + Sqrt[5]/2)^m) *)
For comparison with the numerical solution by Bob Hanlon,
%[[2]] /. m -> 1 // N
(* 0.236068 *)