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I was looking at this step from Wolfram alpha

enter image description here

In the above $W$ is LambertW function which Mathematica calls ProductLog

enter image description here

When I try to obtain this solution in Mathematica 13.1, it can not solve it. Then how did Wolfram Alpha obtain this solution?

eq = Log[1 - y[x]] - Log[y[x]] + 1/y[x] == x + C[1]
Solve[eq, y[x]]

Mathematica graphics

Adding Reals did not help.

To see the solution above produced by Wolfram alpha, you can type this command inside Mathematica

WolframAlpha["solve y'=y^2*(y-1), y(0)=1"]

Then when the result comes back to the notebook, click on the Show steps button on the top right of the result, and now you will see the above solution/step in the middle of the steps shown.

Btw, Mathematica also solved this ODE and gives same answer as Wolfram alpha.

DSolve[{y'[x] == y[x]^2*(y[x] - 1), y[0] == 1}, y[x], x]
(*{{y[x] -> 1}}*)

Which is correct solution. So it must have done same step internally. But the question why does the Solve command not work in Mathematica? Is there a work around to solve this in Mathematica as shown in Wolfram Alpha steps?

I also tried Reduce and it also could not solve it. Is Wolfram Alpha getting smarter than Wolfram Mathematica?

V 13.1 on windows

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    $\begingroup$ If I combine the Log by hand, Solve[Log[(1-y)/y]+1/y==x+c,y], I get ProductLog in V12.3. $\endgroup$
    – user293787
    Commented Nov 8, 2022 at 6:11
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    $\begingroup$ @user293787 you are right! But I would have expected Mathematica to do this then. It looks like Wolfram Alpha is getting smarter than Wolfram Mathematica these days. May be because it uses A.I. $\endgroup$
    – Nasser
    Commented Nov 8, 2022 at 6:13
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    $\begingroup$ Personally I think Mathematica is conservative when dealing with branch cut. $\endgroup$
    – Lacia
    Commented Nov 8, 2022 at 13:12

1 Answer 1

Reset to default
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$Version

(* "13.1.0 for Mac OS X x86 (64-bit) (June 16, 2022)" *)

Clear["Global`*"]

eq = Log[1 - y[x]] - Log[y[x]] + 1/y[x] == x + C[1];

Off[Solve::ifun]

(sol = Solve[ApplySides[Exp, eq], y[x]][[1]]) // TraditionalForm

enter image description here

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  • $\begingroup$ Do you think this is something that Mathematica should have done automatically? Should I report this? $\endgroup$
    – Nasser
    Commented Nov 8, 2022 at 6:24
  • $\begingroup$ @Nasser - The solution throws an inverse function warning. Presumably, it balked at this and left it to the user to decide whether to proceed. It didn't give an erroneous result so I wouldn't consider it worth reporting. $\endgroup$
    – Bob Hanlon
    Commented Nov 8, 2022 at 6:30

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