If $x \sin \left(\frac{y}{x}\right) dy = \left[y \sin \left(\frac{y}{x}\right) - x\right] dx$,$x > 0$ and $y(1) = \frac{\pi}{2}$,then the value of $\cos \left(\frac{y}{x}\right)$ is

Show that the differential equation $x \cos \left(\frac{y}{x}\right) \frac{d y}{d x}=y \cos \left(\frac{y}{x}\right)+x$ is homogeneous and solve it.

Which of the following is a homogeneous differential equation?

$3 P(A) = P(B) = \frac{5}{13}$ and $P(A \mid B) = \frac{3}{5}$,then $P(A \cup B) = $ . . . . . . .

The solution of the equation $\frac{dy}{dx} = \frac{y}{x} \left( \log \frac{y}{x} + 1 \right)$ is

Let $y=y(x)$ be the solution of the differential equation $\left((x+2) e^{\left(\frac{y+1}{x+2}\right)}+(y+1)\right) d x=(x+2) d y$ with the initial condition $y(1)=1$. If the domain of $y=y(x)$ is an open interval $(\alpha, \beta)$,then $|\alpha+\beta|$ is equal to $......$