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If I have the following equation

(y - 1) ^5 == 32

I can select the real solution with

Solve[(y - 1) ^5 == 32, x, Reals]

Now, I have the following differential equation

y'[x] == Cos[x] / (y[x] - 1) ^4

and the implicit solution should be

(y[x] - 1) ^5 == 5 Sin[x] + 5 C[1])

DSolve gives the five complex solutions in the form

y[x] -> Root[...]

How can I ask only for the solution?

y[x] -> 1 + (5 Sin[x] + 5 C[1]) ^ (1/5)
enzotib
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  • A first order ode (satisfying the Lipschitz condition) may have only one solution to the Cauchy problem. Whenever writing (5 Sin[x] + 5 C[1]) ^ (1/5) the system doesn't know which one of 5 solutions is appropriate and for certain initial conditions there are no solutions. – Artes Jun 22 '20 at 10:50
  • @Artes: for what I know, Mathematica gives a well defined meaning to x^(1/5): it is the root with the greater real part, and between these, the one with greater imaginary part. – enzotib Jun 22 '20 at 10:57
  • No, it doesn't. Examine e.g. this post Finding real roots of negative numbers. I guess that your question cannot be answered reasonably in this form unless you make precise statements. Mathematica provides 5 roots and one of them could be an appropriate solution to the Cauchy problem. This is non-trivial problem, since in general there are no exact radicals to fifth order polynomials. – Artes Jun 22 '20 at 11:45
  • @Artes: it remains the fact that something that can easily be done by hand (for C[1]>=1 the solution that I wrote exists and is a real function, while the others four are always complex, assuming x and C[1] real), cannot be automated by Mathematica. – enzotib Jun 22 '20 at 12:24
  • I think that formulating the problem this way: find the uniqe (explicit) solution to this problem DSolve[y'[x] == Cos[x]/(y[x] - 1)^4 && y[0] == 0, y[x], x] is well defined and you could expect a correct solution. You might also reformulate the problem to make (5 Sin[x] + 5 C[1]) ^ (1/5) a solution. This needs finding appropriate restriction for C[1]. – Artes Jun 22 '20 at 12:33

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