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

Let $y=y(x)$ be the solution of the differential equation $(1+y^2) e^{\tan x} dx + \cos^2 x(1+e^{2 \tan x}) dy = 0$,with $y(0)=1$. Then $y(\frac{\pi}{4})$ is equal to:

If $\left( \frac{2 + \sin x}{1 + y} \right) \frac{dy}{dx} = - \cos x$ and $y(0) = 1$,then $y\left( \frac{\pi}{2} \right) = $

Let a curve $y=f(x)$ pass through the points $(0,5)$ and $(\log_e 2, k)$. If the curve satisfies the differential equation $2(3+y) e^{2x} dx - (7+e^{2x}) dy = 0$,then $k$ is equal to

Given that the slope of the tangent to a curve $y=y(x)$ at any point $(x, y)$ is $\frac{2y}{x^2}$. If the curve passes through the centre of the circle $x^2+y^2-2x-2y=0$,then its equation is

The equation of a curve passing through $(1,0)$ and having the slope of the tangent at any point $(x, y)$ of the curve as $\frac{y-1}{x^2+x}$ is