Which graph could be a solution to the logistic differential equation $\frac{d N}{d t}=5 N \cdot\left(1-\frac{N}{8}\right) ?$

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Question

Which graph could be a solution to the logistic differential equation $\frac{d N}{d t}=5 N \cdot\left(1-\frac{N}{8}\right) ?$

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Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ We start with the logistic differential equation: $\frac{dN}{dt} = 5N \left(1 - \frac{N}{8} ight)$ ‖ ‖ step 2 ⋮ This is a first-order nonlinear ordinary differential equation. ‖ ‖ step 3 ⋮ To solve it, we use the method of separation of variables. Rewrite the equation as: $\frac{dN}{N(1 - \frac{N}{8})} = 5 \, dt$ ‖ ‖ step 4 ⋮ Integrate both sides: $\int \frac{dN}{N(1 - \frac{N}{8})} = \int 5 \, dt$ ‖ ‖ step 5 ⋮ The left side can be integrated using partial fractions: $\int \left(\frac{8}{N} + \frac{8}{8 - N} ight) dN = \int 5 \, dt$ ‖ ‖ step 6 ⋮ This gives: $8 \ln|N| - 8 \ln|8 - N| = 5t + C$ ‖ ‖ step 7 ⋮ Simplify and solve for $N$: $\ln\left|\frac{N}{8 - N} ight| = \frac{5t + C}{8}$ ‖ ‖ step 8 ⋮ Exponentiate both sides: $\frac{N}{8 - N} = e^{\frac{5t + C}{8}}$ ‖ ‖ step 9 ⋮ Let $e^{\frac{C}{8}} = K$, then: $\frac{N}{8 - N} = K e^{\frac{5t}{8}}$ ‖ ‖ step 10 ⋮ Solve for $N$: $N = \frac{8K e^{\frac{5t}{8}}}{1 + K e^{\frac{5t}{8}}}$ ‖ ‖ step 11 ⋮ The general solution is: $N(t) = \frac{8 e^{5t}}{C + e^{5t}}$ where $C = \frac{8}{K}$ ‖ ‖ step 12 ⋮ Given the initial condition $N(0) = 1$, solve for $C$: $1 = \frac{8}{C + 1}$ ‖ ‖ step 13 ⋮ This gives $C = 7$, so the specific solution is: $N(t) = \frac{8 e^{5t}}{7 + e^{5t}}$ ‖ ‖ step 14 ⋮ The solution $N(t) = \frac{8 e^{5t}}{7 + e^{5t}}$ approaches 8 as $t o \infty$. ‖ ‖ step 15 ⋮ Therefore, the graph that shows $N$ starting at 1 and leveling off near 8 is the correct one. ‖ ***A*** ∻Key Concept∻ ⚹Logistic Growth⚹ ∻Explanation∻ ⚹The logistic differential equation models population growth that starts exponentially but levels off as it approaches a carrying capacity.⚹◊What is the significance of the logistic differential equation $\frac{d N}{d t}=5 N \cdot\left(1-\frac{N}{8}\right)$ in relation to the behavior of the graphs A, B, C, and D provided by the student?⍭ Generate me a similar question◊

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