Q.What would be the per cent growth or birth rate per individual per hour for the same population mentioned in the previous question (Question 10)?

Exponential Growth Rate

A quantity grows exponentially when its rate of change is proportional to its current size: the more there is, the faster it grows. This differs sharply from linear growth, where a fixed amount is added each step. In exponential growth the quantity multiplies by the same factor over equal time intervals.

The differential equation

Let $y(t)$ be the quantity and $k > 0$ the proportionality constant. The rate law "rate of change proportional to the current amount" becomes

This is a separable equation. Integrating,

where $y_0 = y(0)$ is the starting value. The constant $k$ is the growth rate: a larger $k$ means faster growth. (If $k < 0$, the very same equation describes exponential decay.)

Reading the growth rate

Over each unit of time, $y$ is multiplied by $e^{k}$. So if the quantity doubles every unit of time, then $e^{k} = 2$, giving $k = \log 2$. This is how a stated doubling time is converted into the constant $k$.

Where it appears

Population growth with unlimited resources, money under continuous compound interest, and the early stage of a spreading process all obey $\dfrac{dy}{dt} = ky$, and therefore follow $y = y_0 e^{kt}$. In each case the same single constant $k$ controls how quickly the quantity multiplies.

Exponential growth and decay problems (population growth, radioactive decay, compound interest) are standard application questions in the NCERT Class 12 Differential Equations chapter, and "exponential growth rate formula class 12" is a frequently searched topic ahead of CBSE boards and JEE Main. Recognising the pattern dy/dt = ky quickly is what separates a fast solve from a slow one in exam conditions.