The general solution of $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{x+y+1}{x+y-1}$ is

A radio-active substance has a half-life of h days, then its initial decay rate is given by (where radio-active substance has initial mass $\mathrm{m}_0$)

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

In a certain culture of bacteria, the rate of increase is proportional to the number present. If there are $10^4$ at the end of 3 hours and $4 \cdot 10^4$ at the end of 5 hours, then there were _________ the beginning.