Solving an equation with random coefficients

Question

I have the following equation: $$\dfrac{V}{\Sigma}=k*\arcsin{\left(\dfrac{y}{R}\right)}+hy\sqrt{R^2-y^2}$$ where: $$V=V_0+\epsilon.$$ $$\Sigma=\Sigma_0+\eta$$ $\epsilon$ is a normally distribuited random variable: $\epsilon=\mathcal{N}(0,\sigma_0^2)$ and $\eta=\mathcal{N}(0,\sigma_1^2)$ I'm stuck on solving the equation respect to the variable $y$. $k\in\mathbb{R}$,$h\in\mathbb{R}$,$R\in\mathbb{R}$

Answer 1

Removing unnecessary constants, the equation can be rewritten

$$\alpha\arcsin z+\beta z\sqrt{1-z^2}=1.$$ (Notice that there are only two independent coefficients.)

As the unknown appears both inside and outside a transcendental function, there is no closed-form solution and you must resort to numerical methods.

Here is what the curves look like for a few values of the coefficients. Depending on these, there can be one or two real solutions (intersect with horizontals).

Fortunately, when computing the zeroes of the derivative, the $\arcsin$ disappears and you end-up with a simple quadratic expression. Hence you can easily discuss and isolate the roots.