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

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 . . . . . . . . .

If $\int_{a}^{x} t y(t) dt = x^2 + y(x)$,then $y$ as a function of $x$ is:

The solution of the differential equation $\frac{dy}{dx} + \frac{y}{x \log_{e} x} = \frac{1}{x}$ under the condition $y = 1$ when $x = e$ is

The solution of the differential equation $y^{\prime} = \frac{1}{e^y - x}$ is

The solution of the differential equation $\frac{dy}{dx} - 2\frac{y}{x} = x^3$ is: