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How do I find the root of a non-integer "polynomial" equation with an unspecified exponent?

I'm trying to solve for theta in terms of the other parameters of this equation:

$$ \theta^{\frac{1}{1-\alpha}}-\theta\Bigl(\frac{m+\epsilon-pv}{\alpha}\Bigr)^{\frac{\alpha}{1-\alpha}}\frac{1}{1-\alpha}= \text{big equation that doesn't involve $\theta$} $$

where $ m, \theta , \epsilon, and \; pv $ are positive values and $ \alpha \in (0,1)$

The previously suggested solutions for equations with non-integer exponents all involve numeric values rather than unknown parameters as is the case here. I've tried solving it algebraically and I've tried using a Taylor Expansion but neither approach got me anywhere. I am out of my depth here and would appreciate any help!

egreg
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Hugo André
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  • It seems you have exponent $r = 1/(1-\alpha) > 1,$ then other constants $B,C,$ with $$ \theta^r - B \theta = C $$ – Will Jagy Oct 30 '18 at 23:15
  • Assuming that interpretation is correct, i know of nothing you can do ahead of time, nothing worth doing with the symbols; at some point, when you can correctly evaluate $r,B,C$ you can find $\theta$ numerically or confirm that there are no real roots. – Will Jagy Oct 30 '18 at 23:21
  • There are series solutions, e.g. in powers of $B C^{1/r - 1}$. – Robert Israel Oct 30 '18 at 23:56
  • e.g. here – Robert Israel Oct 31 '18 at 03:03
  • @WillJagy I see. The problem is that the sought after theta is a threshold value in an economic model that I'm working on. Since it's part of a model I need to express it in terms of other parameters (such as alpha) to show how the threshold changes when they do. – Hugo André Oct 31 '18 at 03:04
  • @RobertIsrael Thank you, that looks perfect! – Hugo André Oct 31 '18 at 03:09

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