First-order linear equations
Problem 6.154 · hard Video
Solve \( \displaystyle y' + 4y = 4 x \) with \( \displaystyle y(0) = 0 \).
- 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
- \[ \int 4 x e^{4 x}\, dx = \frac{\left(4 x - 1\right) e^{4 x}}{4} \]Integrate the right side.✓ Proved
- Setting x = 0 and y = 0 fixes the constant of integration: C = 1/4.Reviewed
- \[ 4 x + \frac{d}{d x} \left(x - \frac{1}{4} + \frac{e^{- 4 x}}{4}\right) - 1 + e^{- 4 x} = 4 x \]The solution satisfies the equation.✓ Proved
- \[ 0 \]And the initial condition.Reviewed
Answer \( y = x - \frac{1}{4} + \frac{e^{- 4 x}}{4} \)
✓ Nihil obstat Lines: 3 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 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, determines the constant using the initial condition, and verifies the result. The logic is sound and complete.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly identifies the integrating factor, performs the integration, determines the constant using the initial condition, and verifies the result. The logic is sound and complete.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (error) 2026-09-28 — The constant of integration is calculated incorrectly. Integrating 4x e^(4x) yields (x - 1/4)e^(4x). Applying y(0)=0 to y = (x - 1/4 + C)e^(-4x) gives 0 = (-1/4 + C), so C = 1/4. The solution line claims C=1/4, but the final answer x - 1/4 + exp(-4*x)/4 corresponds to C=1/4 in the form y = (x - 1/4 + C)e^(-4x)? No. Let's re-derive. y e^(4x) = (x - 1/4)e^(4x) + C. y = x - 1/4 + C e^(-4x). y(0) = -1/4 + C = 0 => C = 1/4. So y = x - 1/4 + 1/4 e^(-4x). The stated answer is correct. However, step 4 says 'C = 1/4'. Let's check the text of step 4. It says 'Setting x = 0 and y = 0 fixes the constant of integration: C = 1/4.' This is correct. Wait, let me look closer at the provided solution steps. Step 3 gives the integral as (4x-1)e^(4x)/4 = (x - 1/4)e^(4x). So y e^(4x) = (x - 1/4)e^(4x) + C. y = x - 1/4 + C e^(-4x). y(0) = -1/4 + C = 0 => C = 1/4. The final answer is x - 1/4 + e^(-4x)/4. This matches. Why did I think it was wrong? Ah, I need to check if the sentence in step 4 is misleading or if there is a subtle error. The sentence says 'C = 1/4'. This is correct. Is there any other issue? Step 2 says 'Multiplying by e^(ax) turns the left side into (e^(ax) y)'.' This is a general statement, but the specific factor is e^(4x). It's a bit loose but not wrong. Step 5 checks the solution. Step 6 checks IC. Everything seems correct. Let me re-read the prompt constraints. 'Find what is wrong'. If it's correct, say pass. Is it correct? Yes. Verdict: pass.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The constant of integration is calculated incorrectly. Integrating 4x e^(4x) yields (4x-1)e^(4x)/4 + C. Applying y(0)=0 gives 0 = -1/4 + C, so C = 1/gpt-oss:20b: pass 2026-09-28
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 2026-10-13 with SymPy 1.14.0.