The logistic equation
Problem 6.343 · hard
A population follows the logistic model with growth rate \( \displaystyle r = \frac{1}{2} \) per year, carrying capacity 200, and initial population 4. Find \( \displaystyle P(t) \), \( \displaystyle P(7) \), and when the population reaches 100.
- Equilibria: P = 0 and P = K = 200. The solution has the form P = K/(1 + Ae^(−rt)) with A = (K − P₀)/P₀.
- \[ 49 \]A.✓ Proved
- \[ \frac{d}{d t} \frac{200}{1 + 49 e^{- \frac{t}{2}}} = \frac{100 - \frac{100}{1 + 49 e^{- \frac{t}{2}}}}{1 + 49 e^{- \frac{t}{2}}} \]This P(t) satisfies the differential equation.✓ Proved
- \[ 4 \]and the initial condition.✓ Proved
- \[ \frac{200}{1 + \frac{49}{e^{\frac{7}{2}}}} = \frac{200 e^{\frac{7}{2}}}{e^{\frac{7}{2}} + 49} \]P(7).✓ Proved
- \[ 2 \ln{\left(49 \right)} = \ln{\left(2401 \right)} \]Half the carrying capacity when Ae^(−rt) = 1, i.e. t = ln(A)/r.✓ Proved
Answer \( P(t) = \frac{200}{1 + 49 e^{- \frac{t}{2}}},\ P(7) \approx 80.7,\ t = \ln{\left(2401 \right)} \approx 7.784 \)
Lines: 5 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the differential equation solved numerically (Taylor-series ODE solver) from P(0) |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the final answer for P(7) as a numerical approximation, which is requested in the problem statement. Additionally, the final sentence is confusingly phrased, linking the logarithmic identity to the time calculation without clearly stating the final value for t.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution fails to explicitly state the final answer for P(7) as a numerical approximation, which is requested in the problem statement. Additionally, the final sentence is confusingly phrased, linking the logarithmic identity to the time calculation without clearly stating the final value for t.qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly derives the logistic function, verifies the differential equation and initial condition, and accurately computes the specific values requested.gpt-oss:20b: pass 2026-10-10
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/logistic_equation, checked 2026-10-10 with SymPy 1.14.0.