The solution of the differential equation $({e^x} + 1)y \, dy = (y + 1){e^x} \, dx$ is:

The particular solution of the differential equation $(y + x \cdot \frac{dy}{dx}) \cdot \sin(xy) = \cos x$ at $x = 0$ is

Let $S$ be the family of curves given by the general solution of the differential equation $\frac{y^2 e^{-1 / y}}{\sqrt{x}} dx - 2 \sec \sqrt{x} dy = 0$. Then the equation of the curve belonging to $S$ and passing through $(\pi^2, 1)$ is

If the differential equation $\frac{dy}{dx} + \frac{x}{y} = \frac{a}{y}$,where $a$ is a constant,represents a family of circles,then the radius of the circle is ......

Find the equation of a curve passing through the point $(0, -2)$ given that at any point $(x, y)$ on the curve,the product of the slope of its tangent and the $y$-coordinate of the point is equal to the $x$-coordinate of the point.

The general solution of the differential equation $\log \left(\frac{dy}{dx}\right) = ax + by$ is