First-order linear equations
Problem 6.64 · medium Video
Solve \( \displaystyle y' + 4y = e^{- 4 x} \) with \( \displaystyle y(0) = 0 \).
- The equation is linear in standard form; the integrating factor is e^(∫4 dx) = e^(4x).
- \[ \frac{d}{d x} Y{\left(x \right)} e^{4 x} = 4 Y{\left(x \right)} e^{4 x} + e^{4 x} \frac{d}{d x} Y{\left(x \right)} \]Multiplying by e^(ax) turns the left side into (e^(ax) y)'.✓ Proved
- \[ \int 1\, dx = x \]Integrate the right side.✓ Proved
- Setting x = 0 and y = 0 fixes the constant of integration: C = 0.
- \[ 4 x e^{- 4 x} + \frac{d}{d x} x e^{- 4 x} = e^{- 4 x} \]The solution satisfies the equation.✓ Proved
- \[ 0 \]And the initial condition.
Answer \( y = x e^{- 4 x} \)
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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution skips the key step of multiplying the differential equation by the integrating factor and integrating the resulting left‑hand side. It also iqwen3.6:27b-mlx: fail (error) — The solution fails to explicitly derive the general solution y = (x + C)e^{-4x} from the integrated form. It jumps from the integrated right-hand side (x) to the specific solution with C=0 without showing the division by the integrating factor or the determination of C from the general form, making the logical flow 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 + C)e^{-4x} from the integrated form. It jumps from the integrated right-hand side (x) to the specific solution with C=0 without showing the division by the integrating factor or the determination of C from the general form, making the logical flow incomplete.gpt-oss:20b: inconclusive 2026-09-28 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution skips the key step of multiplying the differential equation by the integrating factor and integrating the resulting left‑hand side. It also iqwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the integrating factor, performs the integration, applies the initial condition, and verifies the result. The logic is sound and complete.gpt-oss:20b: fail (misleading) 2026-09-27 — The solution omits the constant of integration when integrating the right‑hand side, then later claims C=0 without having introduced it. This could mislead a student into thinking the constant is unnecessary or that the integration step is trivial.qwen3.6:27b-mlx: pass 2026-09-26 — 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 2026-09-26
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 2026-10-13 with SymPy 1.14.0.