Second-order linear equations
Problem 6.78 · medium
Solve \( \displaystyle y'' + 0y' + 9y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = 0 \).
- Try y = e^(mx): the characteristic equation is m² + bm + c = 0.
- \[ \left(m - 3 i\right) \left(m + 3 i\right) = m^{2} + 9 \]Roots: -3*I, 3*I.✓ Proved
- Roots α ± βi give y = e^(αx)(C₁cos βx + C₂sin βx).
- Use y(0) and y'(0) to find C₁ and C₂.
- \[ 9 \cos{\left(3 x \right)} + \frac{d^{2}}{d x^{2}} \cos{\left(3 x \right)} = 0 \]The solution satisfies the equation.✓ Proved
- \[ 1 \]y(0).
- \[ 0 \]y'(0).
Answer \( y = \cos{\left(3 x \right)} \)
Lines: 2 proved, 5 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 | Not checked | — | a sentence; read, not computed |
| 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 |
| 7 | 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; both initial conditions match |
Reviewers
gpt-oss:20b: fail (error) — The solution never actually determines the constants C₁ and C₂ from the initial conditions. It merely states that they are found and then claims the final function satisfies the differential equation and the initial conditions without showing the calculation that C₁=1 and C₂=0. This omission means the reasoning is incomplete and could mislead a student into thinking the constants were verified when they were not.qwen3.6:27b-mlx: pass — The solution correctly identifies the characteristic equation, roots, and general form. The constants are implicitly determined correctly (C1=1, C2=0) to satisfy the initial conditions, and the final verification steps confirm the result.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the characteristic equation, roots, and general form. The constants are implicitly determined correctly (C1=1, C2=0) to satisfy the initial conditions, and the final verification steps confirm the result.gpt-oss:20b: fail (error) 2026-09-28 — The solution never actually determines the constants C₁ and C₂ from the initial conditions. It merely states that they are found and then claims the final function satisfies the differential equation and the initial conditions without showing the calculation that C₁=1 and C₂=0. This omission means the reasoning is incomplete and could mislead a student into thinking the constants were verified when they were not.qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the characteristic equation, finds the complex roots, applies the general solution formula for complex conjugate roots, and verifies the initial conditions. The steps are logically sound and mathematically correct.gpt-oss:20b: fail (misleading) 2026-09-27 — The solution jumps to y=cos(3x) without showing that the constants C₁=1 and C₂=0 satisfy the initial conditions. A student could be misled into thinking the general solution automatically yields the particular solution without verifying the constants.qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the characteristic equation, finds the complex roots, applies the general solution formula for complex conjugate roots, and verifies the initial conditions. The logic is sound.gpt-oss:20b: fail (misleading) 2026-09-26 — The solution skips the crucial step of determining the constants C₁ and C₂ from the initial conditions. It merely states that the initial conditions are used but never shows the calculation, leaving the reader uncertain how the final solution cos(3x) was obtained.
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/second_order_linear, checked 2026-10-13 with SymPy 1.14.0.