The integrating factor of $x \frac{dy}{dx} - 2y = x^2 + \sin \left( \frac{1}{x^2} \right)$ is

Let $y=y(x), x>1$,be the solution of the differential equation $(x-1) \frac{d y}{d x}+2 x y=\frac{1}{x-1}$,with $y(2)=\frac{1+e^{4}}{2 e^{4}}$. If $y(3)=\frac{e^{\alpha}+1}{\beta e^{\alpha}}$,then the value of $\alpha+\beta$ is equal to

Let $f:(0, \infty) \rightarrow \mathbb{R}$ be a differentiable function such that $f^{\prime}(x)=2-\frac{f(x)}{x}$ for all $x \in(0, \infty)$ and $f(1) \neq 1$. Then

Find the general solution of the differential equation: $(1+x^{2}) dy + 2xy dx = \cot x dx$ where $x \neq 0$.

The solution of $\frac{dy}{dx} + \frac{1}{3}y = 1$ is

The general solution of the differential equation $\frac{dy}{dx} + xy = 4x - 2y + 8$ is