Find the general solution of the differential equation $\frac{dy}{dx}-y=\cos x$.

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

Let $y=y(x)$ be the solution of the differential equation $(2x \ln x) \frac{dy}{dx} + 2y = \frac{3}{x} \ln x$,$x > 0$ and $y(e^{-1}) = 0$. Then,$y(e)$ is equal to

The integrating factor of $y + \frac{d}{dx}(xy) = x(\sin x + \log x)$ is

If $y=y(x)$ is the solution of the differential equation $\frac{dy}{dx}+\frac{4x}{x^2-1}y=\frac{x+2}{(x^2-1)^{5/2}}$ for $x > 1$,such that $y(2)=\frac{2}{9}\log_e(2+\sqrt{3})$ and $y(\sqrt{2})=\alpha\log_e(\sqrt{\alpha}+\beta)+\beta-\sqrt{\gamma}$,where $\alpha, \beta, \gamma \in N$,then $\alpha\beta\gamma$ is equal to $........$.

Find the particular solution of the following differential equation,given that $y=1$ when $x=0$: $(1+x^2) \frac{dy}{dx} = e^{\tan^{-1} x} - y$.