First-order linear equations
Problem 6.21 · hard Video
Solve \( \displaystyle y' + 4y = 6 \) 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
- \[ \int 6 e^{4 x}\, dx = \frac{3 e^{4 x}}{2} \]Integrate the right side.✓ Proved
- Setting x = 0 and y = 0 fixes the constant of integration: C = -3/2.Reviewed
- \[ \frac{d}{d x} \left(\frac{3}{2} - \frac{3 e^{- 4 x}}{2}\right) + 6 - 6 e^{- 4 x} = 6 \]The solution satisfies the equation.✓ Proved
- \[ 0 \]And the initial condition.Reviewed
Answer \( y = \frac{3}{2} - \frac{3 e^{- 4 x}}{2} \)
Lines: 3 proved, 3 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. 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, applies the initial condition to find the constant, and verifies the final result.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the integrating factor, performs the integration, applies the initial condition to find the constant, and verifies the final result.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the integrating factor, performs the integration, and determines the constant of integration to satisfy the initial condition. The final verification steps confirm the result.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 4 claims C = -3/2, but substituting x=0, y=0 into y*e^(4x) = 3/2*e^(4x) + C yields 0 = 3/2 + C, so C = -3/2 is correct. However, the general solution is y = 3/2 + C*e^(-4x). With C=-3/2, y = 3/2 - 3/2*e^(-4x). The stated answer matches. Wait, let me re-read carefully. Step 4 says 'Setting x=0 and y=0 fixes... C=-3/2'. This is correct. Step 5 checks the derivative. The solution is correct. Why did I think it failed? Let me check the integration constant logic again. y e^{4x} = 3/2 e^{4x} + C. y(0)=0 => 0 = 3/2 + C => C = -3/2. y = 3/2 - 3/2 e^{-4x}. This is correct. Is there a subtle error? Step 2 says 'Multiplying by e^(ax) turns the left side into (e^(ax) y)''. This is a standard explanation. Step 3 integrates 6e^{4x} to get 3/2 e^{4x}. Correct. Step 4 calculates C. Correct. Step 5 verifies the ODE. Correct. Step 6 verifies IC. Correct. The solution is sound. I will mark it pass.gpt-oss:20b: pass 2026-09-26
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.