The foci of the curve which satisfies the differential equation $(1 + y^2) dx - xy\, dy = 0$ and passes through the point $(1, 0)$ are:

If $\frac{d^2y}{dx^2} + \sin x = 0$,then the solution of the differential equation is ...... .

The solution of the differential equation $x dy = (y + xy^3 (1 + \log_e x)) dx$ is (where $C$ is an arbitrary constant):

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=1+xe^{y-x}$,where $-\sqrt{2} < x < \sqrt{2}$ and $y(0)=0$. Then,the minimum value of $y(x)$ for $x \in(-\sqrt{2}, \sqrt{2})$ is equal to:

If the curve $y = y(x)$ represented by the solution of the differential equation $(2xy^2 - y)dx + xdy = 0$ passes through the intersection of the lines $2x - 3y = 1$ and $3x + 2y = 8$,then $|y(1)|$ is equal to ...... .

Let $y=y(x)$ be the solution of the differential equation $(x+y+2)^2 dx=dy$,$y(0)=-2$. Let the maximum and minimum values of the function $y=y(x)$ in $\left[0, \frac{\pi}{3}\right]$ be $\alpha$ and $\beta$,respectively. If $(3\alpha+\pi)^2+\beta^2=\gamma+\delta\sqrt{3}$,where $\gamma, \delta \in Z$,then $\gamma+\delta$ equals: