Let $x = x(y)$ be the solution of the differential equation $y = (x - y \frac{dx}{dy}) \sin(\frac{x}{y})$,$y > 0$ and $x(1) = \frac{\pi}{2}$. Then $\cos(x(2))$ is equal to:

If $y=y(x)$ is the solution of the differential equation $\left(\frac{5+e^x}{2+y}\right) \frac{dy}{dx}+e^x=0$ satisfying $y(0)=1$,then a value of $y(\log 13)$ is

If $\frac{dy}{dx} = \frac{xy + y}{xy + x}$,then the solution of the differential equation is:

Let the solution curve of the differential equation $x dy - y dx = \sqrt{x^{2} + y^{2}} dx$, where $x > 0$ and $y(1) = 0$, be $y = y(x)$. Then $y(3)$ is equal to:

The general solution of the differential equation $\frac{x dy - y dx}{y} = 0$ is . . . . . . .

Let $S$ be the family of curves given by the general solution of the differential equation $\frac{y^2 e^{-1 / y}}{\sqrt{x}} dx - 2 \sec \sqrt{x} dy = 0$. Then the equation of the curve belonging to $S$ and passing through $(\pi^2, 1)$ is