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I'm trying to solve an equation of the form:

$ax + bx^{1+c} + d = 0$, where $0 < c < 1$, and the reciprocal of $c$ is not necessarily an integer either.

Mathematica protests to me that it is not up to the task of solving this, and I'd like a general solution rather than FindRoot at a particular value.

I've poked around and there are a couple work-arounds - usually involving some substitution - for solving a non-integer polynomials but (as far as I can tell - because I'm having trouble applying them to this problem) they all only really seem to work if the non-integer polynomial is less than one and this is definitely more than one.

I've come to the point of getting a derivative w.r.t. to one of the parameters of interest with the implicit function theorem, which is fine for my purposes, but obviously getting a general solution would be preferable.

Is there a general solution for this?

  • 2
    Already when the reciprocal of $c$ is an integer there is no expression for the solution(s) in terms of elementary functions.2012-08-24
  • 2
    Nitpick: This is **not** a polynomial unless $c$ is an integer.2012-08-28

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