The logistic equation
Problem 6.382 · hard
Solve \( \displaystyle P' = \frac{3}{10}P\left(1 - \frac{P}{10000}\right) \), \( \displaystyle P(0) = 500 \). Find the equilibrium solutions, \( \displaystyle P(9) \), and the time when the population reaches half its carrying capacity.
- Equilibria: P = 0 and P = K = 10000. The solution has the form P = K/(1 + Ae^(−rt)) with A = (K − P₀)/P₀.Reviewed
- \[ 19 \]A.✓ Proved
- \[ \frac{d}{d t} \frac{10000}{1 + 19 e^{- \frac{3 t}{10}}} = \frac{3000 - \frac{3000}{1 + 19 e^{- \frac{3 t}{10}}}}{1 + 19 e^{- \frac{3 t}{10}}} \]This P(t) satisfies the differential equation.✓ Proved
- \[ 500 \]and the initial condition.✓ Proved
- \[ \frac{10000}{1 + \frac{19}{e^{\frac{27}{10}}}} = \frac{10000 e^{\frac{27}{10}}}{e^{\frac{27}{10}} + 19} \]P(9).✓ Proved
- \[ \frac{10 \ln{\left(19 \right)}}{3} \]Half the carrying capacity when Ae^(−rt) = 1, i.e. t = ln(A)/r.✓ Proved
Answer \( P(t) = \frac{10000}{1 + 19 e^{- \frac{3 t}{10}}},\ P(9) \approx 4391.9,\ t = \frac{10 \ln{\left(19 \right)}}{3} \approx 9.815 \)
✓ Nihil obstat Lines: 5 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly identifies the logistic model parameters, derives the constant A, and solves for the specific time and population values. The logic is sound and the steps are clearly justified.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly identifies the logistic model parameters, derives the constant A, and solves for the specific time and population values. The logic is sound and the steps are clearly justified.gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly identifies the logistic model parameters, derives the specific solution satisfying the initial condition, and accurately computes the requested values for P(9) and the time to reach half the carrying capacity.
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 2026-10-12 with SymPy 1.14.0.