The solution of the differential equation $\frac{dx}{x} + \frac{dy}{y} = 0$ is

Find a particular solution satisfying the given condition: $\cos \left(\frac{dy}{dx}\right) = a$ $(a \in R)$; $y = 1$ when $x = 0$.

Find the equation of a curve passing through the point $(-2, 3),$ given that the slope of the tangent to the curve at any point $(x, y)$ is $\frac{2x}{y^2}$.

If $y=y(x)$ is the solution of the differential equation $\left(\frac{2+\sin x}{y+1}\right) \frac{d y}{d x}+\cos x=0$ with $y(0)=1$, then $y\left(\frac{\pi}{2}\right)$ is equal to

The general solution of the differential equation $\frac{dy}{dx} = \frac{1}{x+y+1}$ is ($k, c$ are arbitrary constants)

Find the general solution of the differential equation $\frac{dy}{dx} = \frac{x+1}{2-y}, (y \neq 2)$.