Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=\frac{4y^3+2yx^2}{3xy^2+x^3}$ with $y(1)=1$. If for some $n \in N$,$y(2) \in [n-1, n)$,then $n$ is equal to...

One ticket is selected at random from $50$ tickets numbered $\{00, 01, 02, \ldots, 49\}$. Then the probability that the sum of the digits on the selected ticket is $8$,given that the product of these digits is zero,is

The general solution of the differential equation $(3x^2-2xy)dy+(y^2-2xy)dx=0$ is

The general solution of the differential equation $\left(1+e^{\frac{x}{y}}\right) dx + \left(1-\frac{x}{y}\right) e^{\frac{x}{y}} dy = 0$ is ($C$ is an arbitrary constant).

If $\sin \left(\frac{y}{x}\right)=\log |x|+\frac{\alpha}{2}$ is the solution of the differential equation $x \cos \left(\frac{y}{x}\right) \frac{d y}{d x}=y \cos \left(\frac{y}{x}\right)+x$ and $y(1)=\frac{\pi}{3}$,then $\alpha^2$ is equal to

Find the particular solution satisfying the given condition:
$x^{2} dy + (xy + y^{2}) dx = 0$; $y = 1$ where $x = 1$.