If $\frac{dy}{dx} + y \tan x = \sin 2x$ and $y(0) = 1$,then $y(\pi)$ is equal to

For a differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$,suppose $f^{\prime}(x)=3 f(x)+\alpha$,where $\alpha \in \mathbb{R}, f(0)=1$ and $\lim _{x \rightarrow-\infty} f(x)=7$. Then $9 f\left(-\log _{e} 3\right)$ is equal to ............

Let $y=y(t)$ be a solution of the differential equation $\frac{dy}{dt}+\alpha y=\gamma e^{-\beta t}$,where $\alpha > 0, \beta > 0$ and $\gamma > 0$. Then $\lim_{t \rightarrow \infty} y(t)$ is:

If $y=y(x)$ is the solution of the differential equation $\frac{dy}{dx} + 2y \tan x = \sin x$ with the condition $y(\frac{\pi}{3}) = 0$,then the maximum value of the function $y(x)$ over $\mathbb{R}$ is equal to:

Let $f$ be a differentiable function such that $f'(x) = 7 - \frac{3}{4} \frac{f(x)}{x}$ for $x > 0$ and $f(1) \neq 4$. Then $\lim_{x \to 0^+} x f\left(\frac{1}{x}\right)$

The general solution of the differential equation $\frac{dy}{dx} + (\sec x \operatorname{cosec} x) y = \cos^2 x$ is