The solution of $(xy \cos xy + \sin xy)dx + x^2 \cos xy \, dy = 0$ is

The solution of the differential equation $\frac{dy}{dx} = \frac{\sin y + e^x}{\ln y - x \cos y}$ is:

$A$ function $y = f(x)$ satisfies the condition $f'(x) \sin x + f(x) \cos x = 1$,where $f(x)$ is bounded as $x \rightarrow 0$. If $I = \int_{0}^{\frac{\pi}{2}} f(x) \, dx$,then:

Statement $-1$: The slope of the tangent at any point $P$ on a parabola,whose axis is the $x$-axis and vertex is at the origin,is inversely proportional to the ordinate of the point $P$.
Statement $-2$: The system of parabolas $y^2 = 4ax$ satisfies a differential equation of degree $1$ and order $1$.

The equation of the curve passing through $(3, 4)$ and satisfying the differential equation $y \left( \frac{dy}{dx} \right)^2 + (x - y) \frac{dy}{dx} - x = 0$ can be:

Let $y$ be the solution of the differential equation $x \frac{dy}{dx} = \frac{y^2}{1 - y \log x}$ satisfying $y(1) = 1$. Then, $y$ satisfies: