Is it possible to solve this equation by hand?

I am working on a physics task, and reduced it to the following equation for $y$: $$\frac{1}{4y^3}-\frac{2}{(y^2+b^2)^{\frac{3}{2}}}=0$$ I handed it to Mathematica, and it gave me two real solutions, $$y_{1,2} = \pm\frac{b}{\sqrt{3}},$$ along with some complex ones. My question is, how can I see this? I mean, how can I solve such an equation by hand? I tried moving stuff around a bit, but all I found was a sextic equation, which I really didn't feel like approaching. Is there any easier way? Does it lower the hassle, if only real solutions are to be found?

Solution 1:

First, get rid of the denominators,

$$\begin{align} \frac{1}{4y^3} - \frac{2}{(y^2+b^2)^{3/2}} &= 0\\ \iff \frac{1}{4y^3} &= \frac{2}{(y^2+b^2)^{3/2}}\\ \iff (y^2+b^2)^{3/2} &= 8y^3. \end{align}$$

Then, raise it to the $2/3$-th power to simplify it, introducing a third root of unity,

$$\begin{align} y^2 + b^2 &= 4\rho y^2\\ \iff b^2 &= (4\rho-1)y^2\\ \iff y &= \pm \frac{b}{\sqrt{4\rho-1}}. \end{align}$$

Choosing $\rho = 1$ as the third root of unity yields the two real solutions. $\rho = e^{\pm 2\pi i/3}$ yields non-real solutions.

Solution 2:

Let $x^2=y^2+b^2$. Then $x=\pm2y$ and you find the result

Solution 3:

Hint

Rearrange terms, so you can get $$ 8y^3 - \left( \sqrt{y^2+b^2} \right )^3 = 0 $$ Now, use the fact that $a^3 - b^3 = (a-b)(a^2+ab+b^2)$ $$ \left( 2y - \sqrt{y^2+b^2} \right )\left( 4y^2 + 2y \sqrt{y^2+b^2} + y^2 + b^2\right ) = 0 $$ Can you take it from here?