First-order linear equations
Problem 6.173 · hard Video
Solve \( \displaystyle y' + 4y = 2 x \) 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
- \[ \int 2 x e^{4 x}\, dx = \frac{\left(4 x - 1\right) e^{4 x}}{8} \]Integrate the right side.✓ Proved
- Setting x = 0 and y = 4 fixes the constant of integration: C = 33/8.
- \[ 2 x + \frac{d}{d x} \left(\frac{x}{2} - \frac{1}{8} + \frac{33 e^{- 4 x}}{8}\right) - \frac{1}{2} + \frac{33 e^{- 4 x}}{2} = 2 x \]The solution satisfies the equation.✓ Proved
- \[ 4 \]And the initial condition.
Answer \( y = \frac{x}{2} - \frac{1}{8} + \frac{33 e^{- 4 x}}{8} \)
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: passqwen3.6:27b-mlx: fail (error) — The solution skips the crucial step of dividing by the integrating factor to isolate y(x), jumping directly from the integrated form to the final answer without showing the algebraic manipulation. While the final answer is correct, the logical gap between step 3 and step 4/5 makes the derivation incomplete and potentially confusing for a student learning the method.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution skips the crucial step of dividing by the integrating factor to isolate y(x), jumping directly from the integrated form to the final answer without showing the algebraic manipulation. While the final answer is correct, the logical gap between step 3 and step 4/5 makes the derivation incomplete and potentially confusing for a student learning the method.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the integrating factor, performs the integration, applies the initial condition, and verifies the result. The logic is sound and complete.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-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 mathematically correct.gpt-oss:20b: fail (error) 2026-09-29 — The integral of 2x e^{4x} is e^{4x}(x−1)/2, not (4x−1)e^{4x}/8. Consequently the constant C and the final solution are incorrect.
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.