The general solution of the differential equation $\frac{d y}{d x}=\cot x \cdot \cot y$ is

The equation of a curve passing through $(1,0)$ and having slope of tangent at any point $(x, y)$ of the curve as $\frac{y-1}{x^2+x}$ is

The differential equation which represents the family of curves $y=c_1 e^{c_2 x}$, where $c_1, c_2$ are arbitrary constants is

The rate of increase of the population of a city is proportional to the population present at that instant. In the period of 40 years the population increased from 30,000 to 40,000 . At any time t the population is $(a)(b)^{\frac{t}{40}}$. Then the values of $a$ and $b$ are respectively