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

The general solution of the differential equation $$x+y \frac{d y}{d x}=\sec \left(x^2+y^2\right)$$ is

The differential equation of all circles which pass through the origin and whose centre lie on Y-axis is

An ice ball melts at the rate which is proportional to the amount of ice at that instant. Half the quantity of ice melts in 20 minutes, $$x_0$$ is the initial quantity of ice. If after 40 minutes the amount of ice left is $$\mathrm{Kx}_0$$, then $$\mathrm{K}=$$