How to solve this (difficult) simple equation in one variable?

Question

I have a rather convoluted equation in one variable that I am trying to solve, in terms of many other parameters.

Let $$w(x) = Ax^2 - Bx^4, \quad A,B > 0$$ and $$\varphi = \arccos\left(\rho A^{-2/3}\right),\quad \rho < 0\mbox{ fixed}.$$ I choose here $\arccos\in [0,\pi)$.

Let $$b = \sqrt{\frac{2A}{3B}}\cos{\frac{\varphi}{3}}$$ and $$\beta = \sqrt{\frac{A}{2}}\sin{\frac{\varphi}{3}} - \sqrt{\frac{A}{6B}}\cos{\frac{\varphi}{3}}.$$

Goal: Solve the following for $A$:

$$ w(b) = w(\beta) + (b - \beta + 1)c, \quad c>0 \mbox{ is fixed.} $$

Just note: $A$ is in $b,\beta, \varphi$ as well.

I would appreciate any help on this matter. Thank you.

Answer 1

There may not be a closed-form solution for $A$ in terms of the other variables. I just ran these equations through Sage Notebook and the result is not pretty (click the link to view the result).