Solve equation x^(0.7)-x^(-0.3)==1 get wrong answer??

I try to solve the following equation:

syms x

solve (x^0.7-x^(-0.3)==1,x)

ans =

                                         2.2806043918081745336771958436901
                                          0.301860807970099685692066514691
But the second root is wrong..... the right side is -1 instead of 1.
What's happening ?

Answer 1

[Roger Stafford]

I think I can account for the extraneous "solution" you received. If you substitute x^(0.1) = y, which can be expressed as x = y^10, the corresponding equation in y becomes

   y^7-1/y^3 = 1

and multiplying by y^3 and transposing gives

   y^10-y^3-1 = 0

Matlab's 'roots' gives ten roots for this equation, but only two of them are real-valued. One of the latter is

   y = -0.88711653267660

which would correspond to

   x = y^10 = +.30186080797010

Unfortunately, this is not a solution to the original equation in x, since the equations in x and y are not perfectly equivalent.

Thus, you have caught matlab's 'solve' in an error. It is a mistake that 'aolve' is likely to make when faced with an equation involving fractional-valued powers of the unknown.

Lesson to be learned: You should always check the solutions that 'solve' offers. Some of them may not actually be valid solutions.

[John D'Errico]

Note that vpasolve returns only the one root, although it can find multiple roots when they are there to be found.