The integrating factor of the differential equation $\cos x \frac{dy}{dx} + y \sin x = 1$ is

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:

For $x \in R$,let $y(x)$ be a solution of the differential equation $(x^2-5) \frac{dy}{dx} - 2xy = -2x(x^2-5)^2$ such that $y(2)=7$. Find the maximum value of $y(x)$.

The integrating factor of the differential equation $(x^2 - 1)\frac{dy}{dx} + 2xy = x$ is

If $y=\sin x+A \cos x$ is the general solution of $\frac{dy}{dx}+f(x)y=\sec x$, then an integrating factor of the differential equation is

The general solution of the differential equation $\frac{dy}{dx} + \frac{y \ln y}{x} = \frac{y(\ln y)^2}{x^2}$ is (where $C$ is an arbitrary constant):