The solution of the differential equation $2x \frac{dy}{dx} - y = 0$,with the initial condition $y(1) = 2$,represents which of the following curves?

Let a curve $y=f(x)$ pass through the point $(2, (\ln 2)^2)$ and have slope $\frac{2y}{x \ln x}$ for all positive real values of $x$. Then the value of $f(e)$ is equal to:

If $y = f(x)$ is a monotonically increasing function such that $(\frac{dy}{dx})^2 = 6 - \frac{dy}{dx}$ and $y(0) = 5$, then $y(3) = \dots$

The particular solution of the differential equation $(1+y^2) dx - xy dy = 0$ with the condition $y(1) = 0$ represents:

The solution of the differential equation $x \cos y \, dy = (x e^x \log x + e^x) \, dx$ is

Let $y=y(x)$ be the solution of the differential equation $\frac{2+\sin x}{y+1} \cdot \frac{dy}{dx} = -\cos x$,where $y > 0$ and $y(0) = 1$. If $y(\pi) = a$ and $\frac{dy}{dx}$ at $x = \pi$ is $b$,then the ordered pair $(a, b)$ is equal to: