Solution of the logistic equation Use separation of variables to show that the solution of the initial value problem
P'(t) = rP (1-P/K), P(0) = P₀
is P(t) = K/((K/P₀ − 1)e⁻ʳᵗ + 1)

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Start with the logistic differential equation given: \(P'(t) = rP \left(1 - \frac{P}{K}\right)\), where \(r\) and \(K\) are constants, and \(P(0) = P_0\) is the initial condition.

Rewrite the differential equation in separable form by dividing both sides by \(P \left(1 - \frac{P}{K}\right)\) and multiplying both sides by \(dt\): \(\frac{dP}{P \left(1 - \frac{P}{K}\right)} = r \, dt\).

Simplify the left side by expressing the denominator as a single fraction: \(P \left(1 - \frac{P}{K}\right) = P \left(\frac{K - P}{K}\right) = \frac{P(K - P)}{K}\), so the integral becomes \(\int \frac{K}{P(K - P)} \, dP = \int r \, dt\).

Use partial fraction decomposition to rewrite \(\frac{K}{P(K - P)}\) as \(\frac{A}{P} + \frac{B}{K - P}\), then solve for constants \(A\) and \(B\). After finding \(A\) and \(B\), integrate both sides: \(\int \left(\frac{A}{P} + \frac{B}{K - P}\right) dP = \int r \, dt\).

After integrating, apply the initial condition \(P(0) = P_0\) to solve for the constant of integration. Finally, solve the resulting equation for \(P(t)\) to obtain the explicit solution: \(P(t) = \frac{K}{\left(\frac{K}{P_0} - 1\right) e^{-rt} + 1}\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Logistic Differential Equation

The logistic differential equation models population growth with a carrying capacity, expressed as P'(t) = rP(1 - P/K). Here, r is the growth rate, K is the maximum population, and P(t) is the population at time t. It describes growth that slows as the population approaches K.

Separation of Variables

Separation of variables is a method to solve differential equations by rewriting them so that each variable and its differential are on opposite sides. For the logistic equation, this involves isolating terms with P on one side and t on the other before integrating both sides.

Initial Value Problem and Integration Constants

An initial value problem specifies the value of the solution at a particular point, here P(0) = P₀. After integrating, the constant of integration is determined using this initial condition to find the particular solution that fits the problem.

Solution of the logistic equation Use separation of variables to show that the solution of the initial value problemP'(t) = rP (1-P/K), P(0) = P₀is P(t) = K/((K/P₀ − 1)e⁻ʳᵗ + 1)