The integrating factor of the differential equation $\frac{dy}{dx} + y \tan x = \sec x$ is . . . . . . .

The integrating factor of the differential equation $\cos x \frac{dy}{dx} + y \sin x = 1$ is

If $\log y$ is an integrating factor of $\frac{dx}{dy} + P(y)x = Q(y)$,then $P(y) =$

Let $f : (0, \infty) \to (2, 20)$ be a twice differentiable function such that $\lim_{x \to \infty} (f(x) + f'(x) + f''(x)) = \lim_{x \to \infty} g(x)$,where $\lim_{x \to \infty} g(x)$ exists and is equal to $5$. Then $\lim_{x \to \infty} (f(x) - g(x))$ is equal to:

The solution of the differential equation $\frac{dy}{dx} - 2y \tan 2x = e^x \sec 2x$ is

Let $y=y(x)$ be the solution of the differential equation $\cos x(3 \sin x+\cos x+3) dy = (1+y \sin x(3 \sin x+\cos x+3)) dx$; $0 \leq x \leq \frac{\pi}{2}, y(0)=0$. Then,$y\left(\frac{\pi}{3}\right)$ is equal to ..... .