The population of a town increases at a rate proportional to the population at that time. If the population increases from forty thousand to eighty thousand in 20 years, then the population in another 40 years will be

A particular solution of $3 \mathrm{e}^x \tan y \mathrm{~d} x+\left(1-\mathrm{e}^x\right) \sec ^2 y \mathrm{~d} y=0$ with $y(1)=\frac{\pi}{4}$ is

The equation of the curve passing through the origin and satisfying the equation $\left(1+x^2\right) \frac{\mathrm{d} y}{\mathrm{~d} x}+2 x y=4 x^2$, is

The differential equation of all circles having their centres on the line $y=5$ and touching ( X -axis) is $\qquad$