Q:

Find parametric equation of path of particle if acted on by force $\mu y^{-2}$ parallel to $y$-axis

A particle is acted on by a force parallel to $y$-axis whose acceleration (always towards the $x$-axis) is $\mu y^{-2}$ and whenn $y=a$, it is projected parallel to the $x$-axis with velocity $\sqrt\frac{2\mu}{a}$. Find the parametric equation of the path of the particle. Here $\mu$ is a constant.
Since there is no acceleration along $x$-axis; $x=\sqrt\frac{2\mu}{a}t$
Now for $y$; I proceeded with $v\frac{dv}{dy}=\frac{-\mu}{y^2}$
I got $\frac{dy}{\sqrt{\frac{1}{y}-\frac{1}{a}}}=\sqrt{2\mu}t$
I am having trouble integrating this. I am unable to obtain $y$ in terms of $t$.

A:

$$\sqrt{2\mu}\:t=\int\frac{dy}{\sqrt{\frac{1}{y}-\frac{1}{a}}}=\int\sqrt{\frac{ay}{a-y}}dy$$
Change of variable :$\quad y=a \sin^2(Y)\quad$ leading to :
$$\sqrt{2\mu}\:t=2a^{3/2}\int \sin^2(Y)dY = a^{3/2}\left(Y-\frac{1}{2}\sin(2Y )\right)+\text{constant}$$
$Y=\sin^{-1}(\sqrt{\frac{y}{a}})\quad$ After simplification : 
$$t(y)=\frac{1}{\sqrt{2\mu}}\left( a^{3/2}\sin^{-1}\left(\sqrt{\frac{y}{a}}\right)-\sqrt{ay(a-y)} \right)+C$$
Do not try to express the inverse function $y(t)$ as a combination of a finite number of standard functions. There is no convenient special function available for a closed form. Use numerical calculus to compute or draw $y(t)$ .
A parametric form:
$$\begin{cases}
t = \frac{a^{3/2}}{\sqrt{2\mu}}\left(Y-\frac{1}{2}\sin(2Y )\right)+c \\
y=a\sin^2(Y)
\end{cases}$$

