First-order linear equations
Problem 6.103 · medium
Solve \( \displaystyle y' + 1y = 3 \) with \( \displaystyle y(0) = -1 \).
- The equation is linear in standard form; the integrating factor is e^(∫1 dx) = e^(1x).Reviewed
- \[ \frac{d}{d x} Y{\left(x \right)} e^{x} = Y{\left(x \right)} e^{x} + e^{x} \frac{d}{d x} Y{\left(x \right)} \]Multiplying by e^(ax) turns the left side into (e^(ax) y)'.✓ Proved
- \[ \int 3 e^{x}\, dx = 3 e^{x} \]Integrate the right side.✓ Proved
- Setting x = 0 and y = -1 fixes the constant of integration: C = -4.Reviewed
- \[ \frac{d}{d x} \left(3 - 4 e^{- x}\right) + 3 - 4 e^{- x} = 3 \]The solution satisfies the equation.✓ Proved
- \[ -1 \]And the initial condition.✓ Proved
Answer \( y = 3 - 4 e^{- x} \)
✓ 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, applies the method of integrating factors, and determines the constant of integration using the initial condition. The final verification confirms the result.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the integrating factor, applies the method of integrating factors, and determines the constant of integration using the initial condition. The final verification confirms the result.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the integrating factor, applies the method of integrating factors, and correctly determines the constant of integration using the initial condition. The final verification steps are also correct.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 2026-09-26 with SymPy 1.14.0.