Question

Soluti​​on 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)

Accepted Answer

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}$$ + 1}.$$

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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Key Concepts

Logistic Differential Equation

Separation of Variables

Initial Value Problem and Integration Constants