The logistic equation
Problem 6.303 · hard
Solve \( \displaystyle P' = \frac{1}{2}P\left(1 - \frac{P}{200}\right) \), \( \displaystyle P(0) = 50 \). Find the equilibrium solutions, \( \displaystyle P(2) \), and the time when the population reaches half its carrying capacity.
- Equilibria: P = 0 and P = K = 200. The solution has the form P = K/(1 + Ae^(−rt)) with A = (K − P₀)/P₀.
- \[ 3 \]A.✓ Proved
- \[ \frac{d}{d t} \frac{200}{1 + 3 e^{- \frac{t}{2}}} = \frac{100 - \frac{100}{1 + 3 e^{- \frac{t}{2}}}}{1 + 3 e^{- \frac{t}{2}}} \]This P(t) satisfies the differential equation.✓ Proved
- \[ 50 \]and the initial condition.✓ Proved
- \[ \frac{200}{1 + \frac{3}{e}} = \frac{200 e}{e + 3} \]P(2).✓ Proved
- \[ 2 \ln{\left(3 \right)} = \ln{\left(9 \right)} \]Half the carrying capacity when Ae^(−rt) = 1, i.e. t = ln(A)/r.✓ Proved
Answer \( P(t) = \frac{200}{1 + 3 e^{- \frac{t}{2}}},\ P(2) \approx 95.1,\ t = \ln{\left(9 \right)} \approx 2.197 \)
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 calculates t = ln(9) for when the population reaches half the carrying capacity, but the correct time is t = 2 ln(3) ≈ 2.197. While ln(
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: inconclusive 2026-10-08 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution calculates t = ln(9) for when the population reaches half the carrying capacity, but the correct time is t = 2 ln(3) ≈ 2.197. While ln(qwen3.6:27b-mlx: inconclusive 2026-10-08 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly calculates P(2) by substituting t=1 into the formula (line 5), likely confusing the time t with the rate parameter 1/2. Addigpt-oss:20b: pass 2026-10-08
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 2026-10-08 with SymPy 1.14.0.