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

The solution of the differential equation $(1 + \cos x)dy = (1 - \cos x)dx$ is

The equation of the curve passing through $(1, 0)$ and having a slope of $\frac{y - 1}{x^2 + x}$ is:

The solution of the differential equation $e^x y dx + e^x dy + x dx = 0$ is

The particular solution of the differential equation $\frac{dy}{dx} = e^{2y} \cos x$,when $y(\frac{\pi}{6}) = 0$ 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.