The general solution of the differential equation $\frac{d y}{d x}=\frac{x+y+1}{2 x+2 y+1}$ is

The general solution of the differential equation $\cos x \sin y \, dx + \sin x \cos y \, dy = 0$ is

If a curve passes through the point $(1,1)$ and at any point $(x, y)$ on the curve,the product of the slope of its tangent and $x$ coordinate of the point is equal to the $y$ coordinate of the point,then the curve also passes through the point

Let $f$ be a non-negative function defined on the interval $[0,1]$. If $\int_0^x \sqrt{1-\left(f^{\prime}(t)\right)^2} dt = \int_0^x f(t) dt$ for $0 \leq x \leq 1$ and $f(0)=0$,then:

The general solution of the differential equation $(x-(x+y) \log (x+y)) dx + x dy = 0$ is

If $\frac{dy}{dx} = y + 3 > 0$ and $y(0) = 2$,then $y(\ln 2)$ is equal to: