The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is 1000 at $\mathrm{t}=0$. The number of bacteria is increased by $20 \%$ in 2 hours. If the population of bacteria is 2000 after $\frac{\mathrm{k}}{\log \left(\frac{6}{5}\right)}$ hours, then $\left(\frac{\mathrm{k}}{\log 2}\right)^2$ is

The general solution of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}=y \tan x-y^2 \sec x$ is

Let $y=y(x)$ be the solution of the differential equation $(x \log x) \frac{d y}{d x}+y=2 x \log x(x \geq 1)$ then $y(\mathrm{e})$ is equal to

If $\frac{\mathrm{d} y}{\mathrm{~d} x}=y+3, y+3>0$ and $y(0)=2$, then $y(\log 2)$ is equal to