First-order linear equations
Problem 6.154 · hard
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).
- \[ \frac{d}{d x} Y{\left(x \right)} e^{4 x} = 4 Y{\left(x \right)} e^{4 x} + e^{4 x} \frac{d}{d x} Y{\left(x \right)} \]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.
- \[ 4 x + \frac{d}{d x} \left(x - \frac{1}{4} + \frac{e^{- 4 x}}{4}\right) - 1 + e^{- 4 x} = 4 x \]The solution satisfies the equation.✓ Proved
- \[ 0 \]And the initial condition.✓ Proved
Answer \( y = x - \frac{1}{4} + \frac{e^{- 4 x}}{4} \)
Lines: 4 proved, 2 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 | Not checked | — | 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: fail (error) — 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.
Every verdict on record (4)
qwen3.6:27b-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 2026-09-28 with SymPy 1.14.0.