Differential Equations Question - None

Question

The population $p(t)$ at time $t$ of a certain mouse species satisfies the differential equation $\frac{d}{dt}p(t) = 0.5p(t) - 450$. If $p(0) = 850$, then the time at which the population becomes zero is:

Accepted Answer

Step 1: Separate variables

The differential equation is: $\frac{dp}{dt} = 0.5p - 450$

Rearrange: $\frac{dp}{0.5p - 450} = dt$

Step 2: Integrate both sides

$\int \frac{dp}{0.5p - 450} = \int dt$

Let $u = 0.5p - 450$, then $du = 0.5\,dp$, so $dp = 2\,du$

$\int \frac{2\,du}{u} = t + C$

$2\ln|u| = t + C$

$2\ln|0.5p - 450| = t + C$

Step 3: Use initial condition p(0) = 850

$2\ln|0.5(850) - 450| = 0 + C$

$2\ln|425 - 450| = C$

$2\ln|-25| = C$

$2\ln(25) = C$

Step 4: Write the particular solution

$2\ln|0.5p - 450| = t + 2\ln(25)$

$\ln|0.5p - 450| = \frac{t}{2} + \ln(25)$

$|0.5p - 450| = e^{t/2} \cdot 25 = 25e^{t/2}$

Step 5: Find when p(t) = 0

Set $p = 0$:

$|0.5(0) - 450| = 25e^{t/2}$

$|-450| = 25e^{t/2}$

$450 = 25e^{t/2}$

$18 = e^{t/2}$

$\ln(18) = \frac{t}{2}$

$t = 2\ln(18)$

∴ Answer: c

Question:

Step-by-Step Solution

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