Second-order initial and boundary value problems
Problem 6.649 · medium
Solve \( \displaystyle 4 y + y'' = 0 \) with \( \displaystyle y(0) = -3,\ y'(0) = -4 \).
- \[ m^{2} + 4 = \left(m - 2 i\right) \left(m + 2 i\right) \]The characteristic equation has complex roots 0 ± 2i.✓ Proved
- So y = C₁cos(2*x) + C₂sin(2*x); the two conditions fix C₁ and C₂.
- \[ - 8 \sin{\left(2 x \right)} - 12 \cos{\left(2 x \right)} + \frac{d^{2}}{d x^{2}} \left(- 2 \sin{\left(2 x \right)} - 3 \cos{\left(2 x \right)}\right) = 0 \]It satisfies the equation ...✓ Proved
- \[ \left. - 2 \sin{\left(2 x \right)} - 3 \cos{\left(2 x \right)} \right|_{\substack{ x=0 }} = -3 \]... y(0) = -3 ...✓ Proved
- \[ \left. \frac{d}{d x} \left(- 2 \sin{\left(2 x \right)} - 3 \cos{\left(2 x \right)}\right) \right|_{\substack{ x=0 }} = -4 \]y′(0) = -4.✓ Proved
Answer \( y = - 2 \sin{\left(2 x \right)} - 3 \cos{\left(2 x \right)} \)
Lines: 4 proved, 1 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ 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 | finite differences of the answer satisfy the equation, and the conditions hold in floating point |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — Line 1 incorrectly states the roots of the characteristic equation are 0 ± 2i; the roots of m^2 + 4 = 0 are ±2i. While the subsequent general solution is correct, the stated roots are mathematically false.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — Line 1 incorrectly states the roots of the characteristic equation are 0 ± 2i; the roots of m^2 + 4 = 0 are ±2i. While the subsequent general solution is correct, the stated roots are mathematically false.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to actually solve for the constants C1 and C2 using the initial conditions; it merely asserts the form of the solution and then verifies that the final answer satisfies the conditions. This skips the logical step of deriving the coefficients from the given data.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to actually solve for the constants C1 and C2 using the initial conditions. It jumps from the general form to the specific answer without showing the derivation (e.g., y(0) = C1 = -3, y'(0) = 2C2 = -4 => C2 = -2). While the final answer is correct, the logical step of determining the constants is missing.gpt-oss:20b: pass 2026-10-15
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_ivp_bvp, checked 2026-10-19 with SymPy 1.14.0.