The particular solution of the differential equation $$y\left(\frac{d x}{d y}\right)=x \log x$$ at $$x=e$$ and $$y=1$$ is

The differential equation obtained from the function $$y=a(x-a)^2$$ is

The differential equation of all lines perpendicular to the line $$5 x+2 y+7=0$$ is

The bacteria increases at the rate proportional to the number of bacteria present. If the original number '$$N$$' doubles in $$4 \mathrm{~h}$$, then the number of bacteria in $$12 \mathrm{~h}$$ will be