The general solution of the differential equation $\frac{dy}{dx} + \frac{y}{x} = x^2 + 5$ is ....

If $y = y(x)$ is the solution of the differential equation $\sqrt{4-x^2} \frac{dy}{dx} = \left(\left(\sin^{-1}\left(\frac{x}{2}\right)\right)^2 - y\right) \sin^{-1}\left(\frac{x}{2}\right)$ for $-2 \leq x \leq 2$ with $y(2) = \frac{\pi^2-8}{4}$,then $y^2(0)$ 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 $y(x)$.

If $\sin x$ is the integrating factor of the linear differential equation $\frac{dy}{dx} + Py = Q$,then $P$ is

The solution of the equation $\frac{dy}{dx} + 2y \tan x = \sin x$ satisfying $y = 0$ when $x = \frac{\pi}{3}$ 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: