If $\sin x$ is the integrating factor of the linear differential equation $\frac{dy}{dx} + Py = Q$,then $P$ is

The general solution of the differential equation $dx = (2x + 3y - 4) dy$ is

If $x = x(y)$ is the solution of the differential equation $y \frac{dx}{dy} = 2x + y^{3}(y+1)e^{y}$ with the initial condition $x(1) = 0$,then $x(e)$ is equal to:

Integrating factor of $x \frac{dy}{dx} - y = x^4 - 3x$ is

If $y(x)$ is the solution of the differential equation $x \log x \frac{dy}{dx} + y = 2x \log x$,then $y(e)$ is equal to

$A$ function $y = f(x)$ satisfying the differential equation $\frac{dy}{dx} \sin x - y \cos x + \frac{\sin^2 x}{x^2} = 0$ is such that $y \rightarrow 0$ as $x \rightarrow \infty$. Which of the following statements is correct?