The slope of a curve at any point is inversely proportional to twice the $y$-coordinate of that point. If the curve passes through $(4, 3)$,then the equation of the curve is:

The solution of $e^{dy/dx} = x+1, y(0) = 3$ is

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

The solution of the differential equation $x \, dy - y \, dx = 0$ represents:

Let a solution $y=y(x)$ of the differential equation $x \sqrt{x^2-1} dy - y \sqrt{y^2-1} dx = 0$ satisfy $y(2) = \frac{2}{\sqrt{3}}$.
$STATEMENT-1$: $y(x) = \sec \left(\sec^{-1} x - \frac{\pi}{6}\right)$
$STATEMENT-2$: $y(x)$ is given by $\frac{1}{y} = \frac{2\sqrt{3}}{x} - \sqrt{1 - \frac{1}{x^2}}$

$A$ differentiable function satisfies $3f^2(x) f'(x) = 2x$. Given $f(2) = 1$,then the value of $f(3)$ is: