The solution of the differential equation $x \cdot \sin \left(\frac{y}{x}\right) dy = \left[y \cdot \sin \left(\frac{y}{x}\right) - x\right] dx$ is

Let $y = y(x)$ be the solution of the differential equation $x \sin(\frac{y}{x}) dy = (y \sin(\frac{y}{x}) - x) dx$, $y(1) = \frac{\pi}{2}$ and let $\alpha = \cos(\frac{e^{12}}{e^{12}})$. Then the number of integral values of $p$, for which the equation $x^2 + y^2 - 2px + 2py + \alpha + 2 = 0$ represents a circle of radius $r \leq 6$, is . . . . . . .

Let the solution curve $y = y(x)$ of the differential equation $\left[\frac{x}{\sqrt{x^{2}-y^{2}}}+e^{\frac{y}{x}}\right] x \frac{dy}{dx} = x + \left[\frac{x}{\sqrt{x^{2}-y^{2}}}+e^{\frac{y}{x}}\right] y$ pass through the points $(1, 0)$ and $(2\alpha, \alpha)$,where $\alpha > 0$. Then $\alpha$ is equal to:

The solution of the differential equation $x dy - y dx = \sqrt{x^2+y^2} dx$, given that $y=1$ when $x=\sqrt{3}$, is

The general solution of the differential equation $x \left( \frac{dy}{dx} \right) = y \ln \left( \frac{y}{x} \right)$ is:

The general solution of the differential equation $(x^2+xy)y'=y^2$ is