First-order linear equations
Problem 6.127 · hard
Solve \( \displaystyle y' + 2y = 2 e^{x} \) with \( \displaystyle y(0) = -3 \).
- The equation is linear in standard form; the integrating factor is e^(∫2 dx) = e^(2x).Reviewed
- \[ \frac{d}{d x} Y{\left(x \right)} e^{2 x} = 2 Y{\left(x \right)} e^{2 x} + e^{2 x} \frac{d}{d x} Y{\left(x \right)} \]Multiplying by e^(ax) turns the left side into (e^(ax) y)'.✓ Proved
- \[ \int 2 e^{3 x}\, dx = \frac{2 e^{3 x}}{3} \]Integrate the right side.✓ Proved
- Setting x = 0 and y = -3 fixes the constant of integration: C = -11/3.Reviewed
- \[ \frac{\left(4 e^{3 x} - 22\right) e^{- 2 x}}{3} + \frac{d}{d x} \frac{\left(2 e^{3 x} - 11\right) e^{- 2 x}}{3} = 2 e^{x} \]The solution satisfies the equation.✓ Proved
- \[ -3 \]And the initial condition.✓ Proved
Answer \( y = \frac{\left(2 e^{3 x} - 11\right) e^{- 2 x}}{3} \)
✓ Nihil obstat Lines: 4 proved, 2 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 | 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, determines the constant of integration using the initial condition, and verifies the final result.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the integrating factor, performs the integration, determines the constant of integration using the initial condition, and verifies the final result.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27 — 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 2026-09-27
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 2026-09-27 with SymPy 1.14.0.