Finding the value of y in terms of x. Is it possible to get the value of $y$ in terms of $x$ from the below equation? If so please give give me a clue how to do that :)
$$y \sqrt{y^2 + 1} + \ln\left(y + \sqrt{y^2 + 1}\right) = \frac{a}{x^2}.$$

Since $\ \rm{asinh}(y)=\ln\left(y + \sqrt{y^2 + 1}\right)\ $ let's set $\ y:=\sinh(u)\ $ and rewrite your equation as :
$$\sinh(u) \sqrt{\sinh(u)^2 + 1} + u = \frac{a}{x^2}$$
$$\sinh(u) \cosh(u) + u = \frac{a}{x^2}$$
$$\sinh(2u) + 2u = 2\frac{a}{x^2}$$
After that I fear you'll have to solve this numerically (to get $u$ in function of $x$).
I don't see something simpler sorry...

To solve $\ \sinh(w) + w = r\ $ numerically you may :


*

*use iterations (Newton-Raphson) : $\ \displaystyle w_{n+1}=w_n-\frac  {\sinh(w_n)+w_n-r}{\cosh(w_n)+1}$
(starting with $w_0=\frac r2$)

*use reversion of series to get $w=w(r)$ :
$w(r)= \frac 12 r - \frac 1{96}r^3 + \frac 1{1920}r^5 - \frac{43}{1290240}r^7 + \frac {223}{92897280}r^9 - \frac{60623}{326998425600}r^{11} + \frac{764783}{51011754393600}r^{13} - \frac {107351407}{85699747381248000}r^{15} + \mathrm{O}\bigl(r^{17}\bigr)$

*perhaps that other methods exist in the litterature...

