Question:medium
The rate of change of population $P(t)$ with respect to time $(t)$, where $\alpha$, $\beta$ are the constant birth and death rates, respectively, is
The rate of change of population $P(t)$ with respect to time $(t)$, where $\alpha$, $\beta$ are the constant birth and death rates, respectively, is
Show Hint
In population models, net growth = birth rate $-$ death rate. Multiply this with current population for total rate of change.
Updated On: Jan 14, 2026
- $\dfrac{dP}{dt} = (\alpha + \beta)P$
- $\dfrac{dP}{dt} = (\alpha - \beta)P$
- $\dfrac{dP}{dt} = \dfrac{\alpha + \beta}{P}$
- $\dfrac{dP}{dt} = \dfrac{\alpha - \beta}{P}$
Show Solution
The Correct Option is B
Solution and Explanation
The population change rate quantifies the net effect of births and deaths relative to the existing population size.
Births increase the population count, whereas deaths decrease it.
Therefore, $\text{Net Growth Rate} = \alpha - \beta$
The increment in population size directly correlates with the current population $P(t)$.
Consequently, $\frac{dP}{dt} = (\alpha - \beta)P$
Births increase the population count, whereas deaths decrease it.
Therefore, $\text{Net Growth Rate} = \alpha - \beta$
The increment in population size directly correlates with the current population $P(t)$.
Consequently, $\frac{dP}{dt} = (\alpha - \beta)P$
Was this answer helpful?
0
Top Questions on Differential Equations
- If $y = f(x)$ is the solution of the differential equation $(1 + \sin x) \frac{dy}{dx} + \cos x = 0$, such that $f(0) = 0$, then $f\left(\frac{\pi}{2}\right)$ is:
- JEE Main - 2026
- Mathematics
- Differential Equations
- Let \( f(x) = \begin{cases} \frac{ax^2 + 2ax + 3}{4x^2 + 4x - 3}, & x \neq -\frac{3}{2}, \frac{1}{2} \\ b, & x = -\frac{3}{2}, \frac{1}{2} \end{cases} \) be continuous at \( x = -\frac{3}{2} \). If \( f(x) = \frac{7}{5} \), then \( x \) is equal to :
- JEE Main - 2026
- Mathematics
- Differential Equations
- Let \( y = y(x) \) be the solution of the differential equation \[ x\frac{dy}{dx} - \sin 2y = x^3(2 - x^3)\cos^2 y,\; x \ne 0. \] If \( y(2) = 0 \), then \( \tan(y(1)) \) is equal to:
- JEE Main - 2026
- Mathematics
- Differential Equations
- If \( y = f(x) \) is the solution of the differential equation \( (1 + \sin x) \frac{dy}{dx} + \cos x = 0 \), such that \( f(0) = 0 \), then \( f\left( \frac{\pi}{2} \right) \) is equal to
- JEE Main - 2026
- Mathematics
- Differential Equations
- Want to practice more? Try solving extra ecology questions todayView All Questions
