Problem 114 - A Complete Resource Book in Mathematics for JEE Main

Question

The population \(p(t)\) at time \(t\) of a certain mouse species satisfies the differential equation \(\frac{d p(t)}{d t}=0.5 p(t)\) \(-450\) with initial condition \(p(0)=850\), then the value of \(t\) for which \(p(t)=0\) is \([2012]\) (A) \(2 \ln 18\) (B) \(\ln 9\) (C) \(\frac{1}{2} \ln 18\) (D) \(\ln 18\)

Step-by-Step Solution

Answer

(A) 2 \ln 18

Step 1: Setup the Differential Equation

The given differential equation is \( \frac{dp(t)}{dt} = 0.5p(t) - 450 \). This is a first-order linear differential equation, and we need to solve it for \( p(t) \).

Step 2: Rearrange the Equation

Rearrange the equation into standard linear form: \( \frac{dp(t)}{dt} - 0.5p(t) = -450 \).

Step 3: Find Integrating Factor

The integrating factor \( \mu(t) \) can be found using \( \mu(t) = e^{\int -0.5 dt} = e^{-0.5t} \).

Step 4: Multiply Through by Integrating Factor

Multiply the entire differential equation by the integrating factor: \( e^{-0.5t} \frac{dp}{dt} - 0.5p(t)e^{-0.5t} = -450e^{-0.5t} \).

Step 5: Recognize the Left Side as Derivative

The left side simplifies to the derivative of \( p(t) e^{-0.5t} \): \( \frac{d}{dt} [p(t) e^{-0.5t}] = -450e^{-0.5t} \).

Step 6: Integrate Both Sides

Integrate both sides: \( \int \frac{d}{dt} [p(t) e^{-0.5t}] dt = \int -450e^{-0.5t} dt \). This results in \( p(t) e^{-0.5t} = 900e^{-0.5t} + C \).

Step 7: Solve for \(p(t)\)

Multiply both sides by \( e^{0.5t} \) to find: \( p(t) = 900 + Ce^{0.5t} \).

Step 8: Apply Initial Conditions

Given \( p(0) = 850 \), substitute to find \( C \): \( 850 = 900 + C \Rightarrow C = -50 \). Thus, \( p(t) = 900 - 50e^{0.5t} \).

Step 9: Solve for \(t\) when \(p(t) = 0\)

Set \( 900 - 50e^{0.5t} = 0 \) and solve for \( t \): \( e^{0.5t} = 18 \). Therefore, \( 0.5t = \ln 18 \) which gives \( t = 2\ln 18 \).

Step 10: Choose the Correct Answer

The value of \( t \) when \( p(t) = 0 \) is \( 2 \ln 18 \), corresponding to option (A).

Key Concepts

Population Dynamics

Population dynamics is a fascinating area of study within ecology that delves into how populations of organisms, like the mouse species in this problem, change over time. It involves examining how birth rates, death rates, and migration affect population size and composition.

In this exercise, we consider a population that follows a differential equation. This equation models the growth of the population of a mouse species under certain conditions. Understanding the population dynamics of this mouse species involves calculating how its numbers evolve depending on its growth rate and external factors, such as resource limitations. This is represented mathematically by the differential equation:
\(\frac{dp(t)}{dt} = 0.5p(t) - 450\)

This form suggests that the population grows at a rate proportional to itself (exponential growth, represented by \(0.5p(t)\)), combined with a constant decrease in rate (represented by \(-450\)). This type of model is often used in situations where predation or resource limits counteract the natural growth potential of a population, thus affecting its density over time.

First-Order Linear Differential Equations

First-order linear differential equations are a type of differential equation that involves the first derivative of a function. These are fundamental in examining a range of real-world phenomena, including electrical circuits, heat transfer, and population dynamics.

For the equation \(\frac{dp(t)}{dt} - 0.5p(t) = -450\) in our problem, we see it's in the standard form:
\(\frac{dy}{dt} + P(t)y = Q(t)\)

Here, \(y = p(t)\) which is the population, \(P(t) = -0.5\), and \(Q(t) = -450\). Such equations are called linear because they do not involve powers, products, or functions of \(y\) other than multiplication by function \(P(t)\). First-order linear differential equations have a characteristic form that allows them to be solved using techniques like the integrating factor method, facilitating the determination of how the population grows over time.

Integrating Factor Method

The integrating factor method is a standard technique to solve first-order linear differential equations. It utilizes a function, known as the integrating factor, to simplify the equation so that both sides can be easily integrated.

For the equation \(\frac{dp(t)}{dt} - 0.5p(t) = -450\), the integrating factor \(\mu(t)\) is calculated as:
\(\mu(t) = e^{\int -0.5 dt} = e^{-0.5t}\)

By multiplying the entire differential equation by this integrating factor, we transform the left side into the derivative of the product of the function that needs solving and the integrating factor.

This results in:
\(\frac{d}{dt} [p(t)e^{-0.5t}] = -450e^{-0.5t}\)

The power of the integrating factor method lies in simplifying the problem into something that we can integrate efficiently on both sides. Once integrated, it allows us to arrive at a solution for the population function \(p(t)\), which can then be used to find specific values as needed, such as determining when the population reaches zero.