The particular solution of the differential equation $x \, dy + 2y \, dx = 0$, given that $y = 1$ when $x = 2$, is:

Let $y=y(x)$ be the solution of the differential equation $dy=e^{\alpha x+y} dx$; $\alpha \in N$. If $y(\log_{e} 2)=\log_{e} 2$ and $y(0)=\log_{e}(\frac{1}{2})$,then the value of $\alpha$ is equal to $.....$

The general solution of the differential equation $\tan x \tan y \, dx + \cos^2 x \operatorname{cosec}^2 y \, dy = 0$ is

The solution of the differential equation $\sec^2 x \tan y \, dx + \sec^2 y \tan x \, dy = 0$ with the initial condition $y(\frac{\pi}{4}) = \frac{\pi}{3}$ is:

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

The general solution of the differential equation $\frac{dy}{dx} = (9x + y + 5)^2$ is...