The solution of $\frac{dy}{dx} + 1 = e^{x+y}$ is

The solution of $\frac{dy}{dx} = \left( \frac{y}{x} \right)^{1/3}$ is

The solution of the differential equation $(1 + x^2)\frac{dy}{dx} = x$ is

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 $\frac{1}{y} = \frac{2\sqrt{3}}{x} - \sqrt{1 - \frac{1}{x^2}}$

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

$A$ curve passing through $(2, 3)$ and satisfying the differential equation $\int\limits_0^x {t\,y(t)\,dt} = x^2y(x)$ for $x > 0$ is