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First-order linear equations

Problem 6.199 · hard

Solve \( \displaystyle y' + 2y = 4 x \) 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
\[ \int 4 x e^{2 x}\, dx = \left(2 x - 1\right) e^{2 x} \]
Integrate the right side.✓ Proved
Setting x = 0 and y = 3 fixes the constant of integration: C = 4.
Reviewed
\[ 4 x + \frac{d}{d x} \left(2 x - 1 + 4 e^{- 2 x}\right) - 2 + 8 e^{- 2 x} = 4 x \]
The solution satisfies the equation.✓ Proved
\[ 3 \]
And the initial condition.✓ Proved

Answer \( y = 2 x - 1 + 4 e^{- 2 x} \)

Lines: 4 proved, 2 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.

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: pass
qwen3.6:27b-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. The steps are logically sound and mathematically correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly identifies the integrating factor, performs the integration, applies the initial condition to find the constant, and verifies the final result. The steps are logically sound and mathematically correct.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly identifies the integrating factor, performs the integration, applies the initial condition to find the constant, and verifies the final result. The steps are logically sound and mathematically correct.
gpt-oss:20b: fail (error) 2026-10-03 — The integral in step 3 is incorrect: ∫4x e^{2x}dx = (2x‑2)e^{2x}+C, not (2x‑1)e^{2x}. Consequently the constant determined in step 4 is wrong and the subsequent solution is invalid.

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 2026-10-03 with SymPy 1.14.0.