The solution of the differential equation $x \frac{dy}{dx} + y = x^3y^6$ is:

The solution of the equation $x\frac{dy}{dx} + 3y = x$ is

Let $y=y(x)$ satisfy the equation $\frac{dy}{dx}-|A|=0$,for all $x>0$,where $A=\begin{bmatrix} y & \sin x & 1 \\ 0 & -1 & 1 \\ 2 & 0 & \frac{1}{x} \end{bmatrix}$. If $y(\pi)=\pi+2$,then the value of $y\left(\frac{\pi}{2}\right)$ is:

Let $y=y(x)$ be the solution of the differential equation $\sec x \frac{dy}{dx} - 2y = 2 + 3 \sin x$, where $x \in (-\frac{\pi}{2}, \frac{\pi}{2})$ and $y(0) = -\frac{7}{4}$. Then $y(\frac{\pi}{6})$ is equal to:

Observe the following statements:
$I$. If $dy+2xy dx=2e^{-x^2} dx$,then $ye^{x^2}=2x+c$
$II$. If $ye^{x^2}-2x=c$,then $dx=\frac{dy}{2e^{-x^2}-2xy}$
Which of the following is a correct statement?

At any point on a curve, the slope of the tangent is equal to the sum of the abscissa and the product of the ordinate and abscissa of that point. If the curve passes through $(0, 1)$, then the equation of the curve is