The logistic equation
Problem 6.263 · hard
A population follows the logistic model with growth rate \( \displaystyle r = \frac{1}{5} \) per year, carrying capacity 10000, and initial population 500. Find \( \displaystyle P(t) \), \( \displaystyle P(5) \), and when the population reaches 5000.
- Equilibria: P = 0 and P = K = 10000. The solution has the form P = K/(1 + Ae^(−rt)) with A = (K − P₀)/P₀.
- \[ 19 \]A.✓ Proved
- \[ \frac{d}{d t} \frac{10000}{1 + 19 e^{- \frac{t}{5}}} = \frac{2000 - \frac{2000}{1 + 19 e^{- \frac{t}{5}}}}{1 + 19 e^{- \frac{t}{5}}} \]This P(t) satisfies the differential equation.✓ Proved
- \[ 500 \]and the initial condition.✓ Proved
- \[ \frac{10000}{1 + \frac{19}{e}} = \frac{10000 e}{e + 19} \]P(5).✓ Proved
- \[ 5 \ln{\left(19 \right)} = \ln{\left(2476099 \right)} \]Half the carrying capacity when Ae^(−rt) = 1, i.e. t = ln(A)/r.✓ Proved
Answer \( P(t) = \frac{10000}{1 + 19 e^{- \frac{t}{5}}},\ P(5) \approx 1251.6,\ t = \ln{\left(2476099 \right)} \approx 14.722 \)
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: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution claims t = ln(2476099) ≈ 14.722, but ln(2476099) ≈ 14.722 is incorrect; ln(2476099) is approximately 14.722 only if the base is e, but
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution claims t = ln(2476099) ≈ 14.722, but ln(2476099) ≈ 14.722 is incorrect; ln(2476099) is approximately 14.722 only if the base is e, butqwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution claims t = ln(2476099) ≈ 14.722, but ln(2476099) ≈ 14.722 is incorrect; ln(2476099) ≈ 14.722 is actually close to 5*ln(19) ≈ 14.722, bugpt-oss:20b: pass 2026-10-06
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 2026-10-06 with SymPy 1.14.0.