First-order linear equations
Problem 6.175 · hard
Solve \( \displaystyle y' + 4y = 2 x \) with \( \displaystyle y(0) = -1 \).
- The equation is linear in standard form; the integrating factor is e^(∫4 dx) = e^(4x).Reviewed
- \[ \frac{d}{d x} Y{\left(x \right)} e^{4 x} = 4 Y{\left(x \right)} e^{4 x} + e^{4 x} \frac{d}{d x} Y{\left(x \right)} \]Multiplying by e^(ax) turns the left side into (e^(ax) y)'.✓ Proved
- \[ \int 2 x e^{4 x}\, dx = \frac{\left(4 x - 1\right) e^{4 x}}{8} \]Integrate the right side.✓ Proved
- Setting x = 0 and y = -1 fixes the constant of integration: C = -7/8.Reviewed
- \[ 2 x + \frac{d}{d x} \left(\frac{x}{2} - \frac{1}{8} - \frac{7 e^{- 4 x}}{8}\right) - \frac{1}{2} - \frac{7 e^{- 4 x}}{2} = 2 x \]The solution satisfies the equation.✓ Proved
- \[ -1 \]And the initial condition.✓ Proved
Answer \( y = \frac{x}{2} - \frac{1}{8} - \frac{7 e^{- 4 x}}{8} \)
Lines: 4 proved, 2 reviewed. 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 | 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 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 | sympy.checkodesol substitutes the solution back; y(0) matches |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the integrating factor, performs the integration, applies the initial condition, and verifies the result. The steps are logically sound and algebraically correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the integrating factor, performs the integration, applies the initial condition, and verifies the result. The steps are logically sound and algebraically correct.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the integrating factor, performs the integration, determines the constant of integration, and verifies the final result. The steps are logically sound and mathematically correct.gpt-oss:20b: fail (error) 2026-09-29 — The integral of 2x e^{4x} was computed incorrectly; the correct antiderivative is e^{4x}(x−1)/2, not (4x−1)e^{4x}/8. Consequently the constant C and the final solution are wrong.
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/linear_first_order, checked 2026-09-29 with SymPy 1.14.0.