A Recurring Problem
A recurring problem I keep running into is solving
$$ax+b=x^c$$As far as I can tell, there is no general closed-form solution for $x$, so I'm maintaining this page as a compilation of things I know about it.
Closed-Form Solutions
The closed-form solutions for $a$ and $b$ are obvious. $x$ has a few closed-form solutions when $c$ is a particular fixed value:
| $c$ | solution |
| $$c=-0.5$$ | some wonky cubic solutions |
| $$c=0$$ | $$ x = \frac{1-b}{a} $$ |
| $$c=0.5$$ | $$ \frac{1-2ab \pm \sqrt{1 - 4ab}}{2a^2} $$ |
| $$c=1$$ | $$ x = \frac{b}{1-a} $$ |
| $$c=2$$ | $$ \frac{a \pm \sqrt{a^2+4b}}{2} $$ |
Responsiveness
We know that
$$ x^c-ax-b=0 $$so
$$ \left( c x^{c-1} - a \right) \delta x - \left( x \right) \delta a - \delta b = 0 $$so
$$ \delta x = \frac{ x \cdot \delta a + \delta b}{c x^{c-1} - a} $$In particular, at $x = 0$, we have
$$ \delta x = \frac{\delta b}{a} $$and at $x = 1$, we have
$$ \frac{\delta a + \delta b}{c - a} $$Other Properties
- If $x=0$, then $b = 0$
- If $x=1$, then $a + b = 1$