The integrating factor of the differential equation $\sin y \left(\frac{d y}{d x}\right) = \cos y (1 - x \cos y)$ is

If $-\frac{\pi}{4} < x < \frac{\pi}{4},$ then the general solution of the differential equation $\cos^{2} x \cdot \frac{dy}{dx} - (\tan 2x) y = \cos^{4} x$ is

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

$A$ 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?

Let $f:[1, \infty) \rightarrow \mathbb{R}$ be a differentiable function. If $6 \int_{1}^{x} f(t) dt = 3xf(x) + x^{3} - 4$ for all $x \ge 1$, then the value of $f(2) - f(3)$ is

An integrating factor of the differential equation $(1 - x^2)\frac{dy}{dx} - xy = 1$ is