A population P grew at the rate given by the equation $$\frac{dP}{dt}=0.5P$$, then the population will become double in

The differential equation of all parabolas whose axis is $$y$$-axis, is

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

The particular solution of the differential equation $$ \frac{d y}{d x}=\frac{x+y+1}{x+y-1} $$ when $$ \mathrm{x}=\frac{2}{3} $$ and $$ y=\frac{1}{3} $$ is