Order and linearity of differential equations
Problem 6.732 · easy
Classify \( \displaystyle - 2 y + y' + 6 = 0 \): its order, whether it is linear and, if so, whether it is homogeneous. Then verify that \( \displaystyle y = e^{2 x} + 3 \) is a solution.
- The highest derivative is y′, so the order is 1.
- It is linear: y and its derivatives appear only to the first power, multiplied by functions of x. The term without y is nonzero: nonhomogeneous.
- \[ - 2 e^{2 x} + \frac{d}{d x} \left(e^{2 x} + 3\right) = 0 \]Substituting y = exp(2*x) + 3 makes the left side 0.✓ Proved
Answer \( \text{order 1, linear and nonhomogeneous} \)
Lines: 1 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. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | Not checked | — | a sentence; read, not computed |
| 3 | ✓ 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 | order from sympy.ode_order; linearity from the equation's second partials in y, y′, y″; the solution checked in floating point |
Reviewers
gpt-oss:20b: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The verification step omitted the constant term +6; the left‑hand side should be \(-2e^{2x}+e^{2x}+6\), which indeed equals 0. The missing term makes theqwen3.6:27b-mlx: pass — The classification of the differential equation is correct, and the verification step correctly substitutes the proposed solution into the equation to show it satisfies the identity.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-25 — The classification of the differential equation is correct, and the verification step correctly substitutes the proposed solution into the equation to show it satisfies the identity.gpt-oss:20b: inconclusive 2026-10-25 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The verification step omitted the constant term +6; the left‑hand side should be \(-2e^{2x}+e^{2x}+6\), which indeed equals 0. The missing term makes theqwen3.6:27b-mlx: pass 2026-10-24 — The classification is correct: the equation is first-order, linear, and nonhomogeneous. The verification step correctly substitutes the proposed solution into the differential equation and confirms it satisfies the equation.gpt-oss:20b: fail (error) 2026-10-24 — The verification step omits the constant term +6 in the differential equation, so the substitution does not correctly check the solution.qwen3.6:27b-mlx: pass 2026-10-24 — The classification of the differential equation is correct, and the verification step correctly substitutes the proposed solution into the original equation to show it satisfies the identity.gpt-oss:20b: fail (error) 2026-10-24 — The verification step omits the constant term +6; substituting y=e^{2x}+3 into the full equation -2y+y'+6=0 yields -2e^{2x}+2e^{2x}+6=6≠0, so the solution is not correct.
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/de_classify, checked 2026-10-25 with SymPy 1.14.0.