First-order linear equations
Problem 6.128 · medium Video
Solve \( \displaystyle y' + 1y = e^{- x} \) with \( \displaystyle y(0) = 2 \).
- 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 = 2 fixes the constant of integration: C = 2.
- \[ \left(x + 2\right) e^{- x} + \frac{d}{d x} \left(x + 2\right) e^{- x} = e^{- x} \]The solution satisfies the equation.✓ Proved
- \[ 2 \]And the initial condition.
Answer \( y = \left(x + 2\right) e^{- x} \)
Lines: 3 proved, 3 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. How we verify
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 | Not checked | — | 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: fail (error) — The solution omits the crucial step of multiplying the right‑hand side by the integrating factor e^x before integrating. Consequently the derivation of the particular integral is incorrect, even though the final expression happens to satisfy the differential equation and the initial condition.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly derive the general solution y(x) = (x + C)e^{-x} before applying the initial condition. While the final answer is correct, the logical step connecting the integrated form to the specific solution with C=2 is missing, making the derivation incomplete.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to explicitly derive the general solution y(x) = (x + C)e^{-x} before applying the initial condition. While the final answer is correct, the logical step connecting the integrated form to the specific solution with C=2 is missing, making the derivation incomplete.gpt-oss:20b: fail (error) 2026-09-28 — The solution omits the crucial step of multiplying the right‑hand side by the integrating factor e^x before integrating. Consequently the derivation of the particular integral is incorrect, even though the final expression happens to satisfy the differential equation and the initial condition.qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the integrating factor, applies the method of integrating factors, and verifies the final result against both the differential equation and the initial condition.gpt-oss:20b: fail (error) 2026-09-27 — Sentence 3 incorrectly states the integral of the right‑hand side; it should integrate e^{-x}, not 1. This misrepresents the step of finding the particular integral.qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the integrating factor, applies the method of integrating factors, and verifies the final result against both the differential equation and the initial condition.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 2026-10-13 with SymPy 1.14.0.