The general solution of the differential equation $\sec y + (x - e^{\sin y}) \frac{dy}{dx} = 0$ is...

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

Let $f:[1, \infty) \rightarrow [2, \infty)$ be a differentiable function such that $f(1)=2$. If $6 \int_1^x f(t) dt = 3x f(x) - x^3$ for all $x \geq 1$,then the value of $f(2)$ is

The general solution of the differential equation $(1+y^2) dx = (\tan^{-1} y - x) dy$ is

If the solution curve $y=f(x)$ of the differential equation $(x^{2}-4)y^{\prime}-2xy+2x(4-x^{2})^{2}=0$ for $x>2$ passes through the point $(3, 15)$, then the local maximum value of $f$ is:

Let $\alpha$ be a non-zero real number. Suppose $f: R \rightarrow R$ is a differentiable function such that $f(0)=2$ and $\lim _{x \rightarrow-\infty} f(x)=1$. If $f^{\prime}(x)=\alpha f(x)+3$ for all $x \in R$,then $f(-\log _e 2)$ is equal to . . . . . . . . .