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First-order linear equations

Problem 6.105 · medium

Solve \( \displaystyle y' + 1y = e^{- x} \) with \( \displaystyle y(0) = -3 \).

The equation is linear in standard form; the integrating factor is e^(∫1 dx) = e^(1x).
\[ \frac{d}{d x} Y{\left(x \right)} e^{x} = Y{\left(x \right)} e^{x} + e^{x} \frac{d}{d x} Y{\left(x \right)} \]
Multiplying by e^(ax) turns the left side into (e^(ax) y)'.✓ Proved
\[ \int 1\, dx = x \]
Integrate the right side.✓ Proved
Setting x = 0 and y = -3 fixes the constant of integration: C = -3.
\[ \left(x - 3\right) e^{- x} + \frac{d}{d x} \left(x - 3\right) e^{- x} = e^{- x} \]
The solution satisfies the equation.✓ Proved
\[ -3 \]
And the initial condition.✓ Proved

Answer \( y = \left(x - 3\right) e^{- x} \)

Lines: 4 proved, 2 not checked. 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	Not checked	—	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	Not checked	—	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: fail (error) — The solution fails to explicitly derive the general solution Y(x) = (x + C)e^{-x} from the integrated equation. It jumps from the integral of the RHS to the specific constant C without showing the intermediate step of dividing by the integrating factor, making the logical flow incomplete and potentially confusing for a student.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to explicitly derive the general solution Y(x) = (x + C)e^{-x} from the integrated equation. It jumps from the integral of the RHS to the specific constant C without showing the intermediate step of dividing by the integrating factor, making the logical flow incomplete and potentially confusing for a student.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-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 to satisfy the initial condition.
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.