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First-order linear equations

Problem 6.175 · hard Video

Solve \( \displaystyle y' + 4y = 2 x \) with \( \displaystyle y(0) = -1 \).

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
\[ \int 2 x e^{4 x}\, dx = \frac{\left(4 x - 1\right) e^{4 x}}{8} \]
Integrate the right side.✓ Proved
Setting x = 0 and y = -1 fixes the constant of integration: C = -7/8.
Reviewed
\[ 2 x + \frac{d}{d x} \left(\frac{x}{2} - \frac{1}{8} - \frac{7 e^{- 4 x}}{8}\right) - \frac{1}{2} - \frac{7 e^{- 4 x}}{2} = 2 x \]
The solution satisfies the equation.✓ Proved
\[ -1 \]
And the initial condition.Reviewed

Answer \( y = \frac{x}{2} - \frac{1}{8} - \frac{7 e^{- 4 x}}{8} \)

The slope field of y' = 2x - 4y: short segments showing the slope at each point. Several solution curves follow the field; the one through (0, -1) is highlighted, y = x/2 - 1/8 - 7e^(-4x)/8.

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: 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.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-04 — 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 2026-10-04
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the integrating factor, performs the integration, applies the initial condition, and verifies the result. The steps are logically sound and algebraically correct.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the integrating factor, performs the integration, determines the constant of integration, and verifies the final result. The steps are logically sound and mathematically correct.
gpt-oss:20b: fail (error) 2026-09-29 — The integral of 2x e^{4x} was computed incorrectly; the correct antiderivative is e^{4x}(x−1)/2, not (4x−1)e^{4x}/8. Consequently the constant C and the final solution are wrong.

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.