Let $y=y(x)$ be the solution of the differential equation $(xy-5x^2\sqrt{1+x^2})dx+(1+x^2)dy=0$,with $y(0)=0$. Then $y(\sqrt{3})$ is equal to

The general solution of the differential equation $\frac{dy}{dx} + y g'(x) = g(x) g'(x)$ is

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 $y(x)$.

Let $y=y(x)$ be the solution of the differential equation $2 \cos x \frac{d y}{d x}=\sin 2 x-4 y \sin x$,where $x \in \left(0, \frac{\pi}{2}\right)$. If $y\left(\frac{\pi}{3}\right)=0$,then $y^{\prime}\left(\frac{\pi}{4}\right)+y\left(\frac{\pi}{4}\right)$ is equal to:

The integrating factor of the differential equation $(1+y+x^{2}y)dx + (x+x^{3})dy = 0$ is

The general solution of the differential equation $(1+\sin^2 x) \frac{dy}{dx} + y \sin 2x = \cos x + \sin^2 x \cos x$ is