First-order linear equations
Problem 6.66 · hard Video
Solve \( \displaystyle y' + 4y = e^{x} \) with \( \displaystyle y(0) = 4 \).
- 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
- \[ \int e^{5 x}\, dx = \frac{e^{5 x}}{5} \]Integrate the right side.✓ Proved
- Setting x = 0 and y = 4 fixes the constant of integration: C = 19/5.
- \[ \frac{\left(4 e^{5 x} + 76\right) e^{- 4 x}}{5} + \frac{d}{d x} \frac{\left(e^{5 x} + 19\right) e^{- 4 x}}{5} = e^{x} \]The solution satisfies the equation.✓ Proved
- \[ 4 \]And the initial condition.
Answer \( y = \frac{\left(e^{5 x} + 19\right) e^{- 4 x}}{5} \)
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 constant of integration is incorrectly computed; plugging x=0 into the general solution gives C=19, not 19/5.qwen3.6:27b-mlx: pass — The solution correctly identifies the integrating factor, performs the integration, applies the initial condition accurately, and verifies the final result.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the integrating factor, performs the integration, applies the initial condition accurately, and verifies the final result.gpt-oss:20b: fail (error) 2026-09-28 — The constant of integration is incorrectly computed; plugging x=0 into the general solution gives C=19, not 19/5.qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the integrating factor, performs the integration, applies the initial condition accurately, and verifies the final result.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the integrating factor, performs the integration, and determines the constant of integration. The final answer matches the stated answer and satisfies both the differential equation and the initial condition.gpt-oss:20b: fail (error) 2026-09-26 — The constant of integration is mis‑computed: from y(0)=4 one gets C=19, not 19/5. The subsequent equations use the correct constant, but the sentence claiming C=19/5 is false.
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.