Let a differentiable function $f$ satisfy $f(x) + \int_{3}^{x} \frac{f(t)}{t} dt = \sqrt{x+1}$ for $x \geq 3$. Then $12f(8)$ is equal to:

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

Let $y=y(x), y>0$,be a solution curve of the differential equation $(1+x^2) dy = y(x-y) dx$. If $y(0)=1$ and $y(2\sqrt{2})=\beta$,then

Let $f$ be a differentiable function such that $x^2 f(x) - x = 4 \int_0^x t f(t) dt$ and $f(1) = \frac{2}{3}$. Then $18 f(3)$ is equal to $......$.

Suppose $y=y(x)$ is the solution curve to the differential equation $\frac{dy}{dx}-y=2-e^{-x}$ such that $\lim_{x \rightarrow \infty} y(x)$ is finite. If $a$ and $b$ are respectively the $x$- and $y$-intercepts of the tangent to the curve at $x=0$,then the value of $a-4b$ is equal to:

The general solution of the differential equation $dx = (2x + 3y - 4) dy$ is