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

If $y = y(x)$ is the solution of the differential equation $\frac{dy}{dx} + (\tan x)y = \sin x, 0 \leq x \leq \frac{\pi}{3},$ with $y(0) = 0,$ then $y\left(\frac{\pi}{4}\right)$ is equal to:

The solution of the differential equation $ydx - xdy + \log x dx = 0$ is

If a curve $y = f(x)$ passes through the point $(1, 2)$ and satisfies $x \frac{dy}{dx} + y = bx^4$,then for what value of $b$ is $\int_{1}^{2} f(x) dx = \frac{62}{5}$?

The general solution of the differential equation $(1+y^{2})+(x-e^{\tan ^{-1} y}) \frac{dy}{dx}=0$ is

Let $y=y(x)$ be the solution of the differential equation $\operatorname{cosec}^{2} x \, dy + 2 \, dx = (1+y \cos 2x) \operatorname{cosec}^{2} x \, dx$,with $y(\frac{\pi}{4})=0$. Then,the value of $(y(0)+1)^{2}$ is equal to: