The population P = P(t) at time 't' of a certain species follows the differential equation dP/dt = 0

Q: Wait, why do we use the absolute value $\ln |P - 900|$? Can't we just write $\ln(P - 900)$?

That is a fantastic question! In indefinite integration, $\int \frac{1}{x} dx = \ln |x| + C$. When we have $P - 900$, if $P < 900$, then $P - 900$ is negative. Since the logarithm of a negative number is undefined in the real number system, the absolute value bars are mathematically mandatory to ensure the domain is valid. Always keep them until you are certain of the sign!

Q: Why can't we just use definite integration from $t=0$ to $t=T$?

You absolutely can! In fact, for JEE Advanced, using definite integrals is often faster and less prone to errors with the constant $C$. You would set the integral from $P=850$ to $P=0$ on the left, and from $t=0$ to $t=T$ on the right. It yields the exact same result and saves you the step of solving for $C$.

Q: Is it possible to solve this using the integrating factor method?

Yes, it is! You could rewrite the equation as $\frac{dP}{dt} - 0.5P = -450$. This is a linear first-order differential equation of the form $\frac{dP}{dt} + P(t)P = Q(t)$. The integrating factor would be $e^{\int -0.5 dt} = e^{-0.5t}$. It works perfectly, but separation of variables is much faster for this specific structure. Choose the path of least resistance!

Question

The population P = P(t) at time 't' of a certain species follows the differential equation $\frac{dP}{dt} = 0.5P - 450$. If $P(0) = 850$, then the time at which population becomes zero is :

Options:

(A) $\frac{1}{2} \log_e 18$

(B) $2 \log_e 18$

(C) $\log_e 9$

(D) $\log_e 18$

Sigma Percentile · Accepted Answer

### The Symphony of Population Dynamics: A Journey into Differential Equations

My dear student, welcome to the fascinating world of population dynamics. Today, we are not just solving a differential equation; we are telling the story of a species.

We are looking at a system that is changing, evolving, and ultimately, fading away. When you look at an equation like $\frac{dP}{dt} = 0.5P - 450$, do not just see symbols. See a living, breathing process.

This equation tells us that the rate at which the population changes is tied directly to its current size. This is the essence of modeling—capturing the heartbeat of reality in the language of mathematics.

### Phase 1: The Anatomy of the Equation

We start with the given differential equation: $\frac{dP}{dt} = 0.5P - 450$. Our goal is to find the time $t$ when the population $P$ hits zero.

Before we rush into the calculus, let us pause and look at the structure. If we factor out the $0.5$, we get:

$$\frac{dP}{dt} = 0.5(P - 900)$$

This is a profound moment. It reveals the 'equilibrium' of the system. If the population were exactly $900$, the rate of change would be zero, and the population would be stable.

However, our initial population is $P(0) = 850$. Since $850 < 900$, the term $(P - 900)$ is negative. This means $\frac{dP}{dt}$ is negative, and the population is destined to decline. We are witnessing a countdown.

### Phase 2: The Art of Separation

In the JEE Advanced arena, the most powerful tool for first-order differential equations is the method of separation of variables. We want to isolate the '$P$-world' from the '$t$-world'.

We take our equation $\frac{dP}{dt} = 0.5(P - 900)$ and perform a mathematical divorce. We move all terms involving $P$ to the left and all terms involving $t$ to the right:

$$\frac{dP}{P - 900} = 0.5 dt$$

Notice the elegance here. We have successfully separated the variables. The left side is now a function of $P$ alone, and the right side is a function of $t$ alone. This is the gateway to integration.

### Phase 3: The Integration

Now, we apply the integral operator to both sides. We are essentially summing up all the infinitesimal changes to find the total behavior of the system:

$$\int \frac{dP}{P - 900} = \int 0.5 dt$$

On the left, we have the standard integral of the form $\int \frac{1}{u} du$, which yields $\ln |u|$. So, we get $\ln |P - 900|$.

On the right, the integral of a constant $0.5$ with respect to $t$ is simply $0.5t$. We must add the constant of integration, $C$, which acts as the 'memory' of our system:

$$\ln |P - 900| = 0.5t + C$$

### Phase 4: The Fingerprint of the System

We know that at $t = 0$, the population $P = 850$. This is our initial condition, our anchor in time. We substitute these values into our equation to find $C$:

$$\ln |850 - 900| = 0.5(0) + C$$

$$\ln |-50| = C$$

$$C = \ln 50$$

With $C$ found, our equation is now fully defined. It is no longer a general solution; it is the specific solution for this exact population:

$$\ln |P - 900| = 0.5t + \ln 50$$

### Phase 5: The Final Countdown

We are at the finish line. The problem asks for the time $t$ when the population becomes zero. So, we set $P = 0$:

$$\ln |0 - 900| = 0.5t + \ln 50$$

$$\ln 900 = 0.5t + \ln 50$$

Now, we isolate $t$. We bring the $\ln 50$ to the left side:

$$0.5t = \ln 900 - \ln 50$$

Using the properties of logarithms, specifically $\ln A - \ln B = \ln(\frac{A}{B})$, we simplify this expression:

$$0.5t = \ln \left( \frac{900}{50} \right)$$

$$0.5t = \ln 18$$

Since $0.5$ is $\frac{1}{2}$, multiplying both sides by $2$ gives us the final answer:

$$t = 2 \ln 18$$

And there it is. The population reaches zero at exactly **$t = 2 \ln 18$**. You have navigated the differential equation, respected the initial conditions, and utilized the properties of logarithms to reach the truth.

Animated Solution for Mathematics - Differential Equations: The population P = P(t) at time 't' of a certain species follows the differential equation $\frac{d P}{d t} = 0.5 P - 450$ . If $P (0) = 850$ , then the time at which population becomes zero is :

Select Answer:

Visualized Solution

Given Differential Equation

Factoring the Equation

Separating the Variables

Integrating Both Sides

Performing the Integration

Applying Initial Condition

Calculating the Constant $C$

The Particular Solution

Setting Population to Zero

Substituting $P = 0$

Rearranging for Time $t$

Using Logarithm Properties

Final Answer

The Sigma Insight: Variable Separable Method

The Symphony of Population Dynamics

Phase 1

Phase 2

Phase 3

Phase 4

Phase 5

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