Second-order initial and boundary value problems
Problem 6.709 · medium
Solve \( \displaystyle 4 y - 4 y' + y'' = 0 \) with \( \displaystyle y(0) = 2,\ y'(0) = -3 \).
- \[ m^{2} - 4 m + 4 = \left(m - 2\right)^{2} \]The characteristic equation has a repeated root 2.✓ Proved
- So y = C₁exp(2*x) + C₂x*exp(2*x); the two conditions fix C₁ and C₂.
- \[ 4 \left(2 - 7 x\right) e^{2 x} - 4 \frac{d}{d x} \left(2 - 7 x\right) e^{2 x} + \frac{d^{2}}{d x^{2}} \left(2 - 7 x\right) e^{2 x} = 0 \]It satisfies the equation ...✓ Proved
- \[ \left. \left(2 - 7 x\right) e^{2 x} \right|_{\substack{ x=0 }} = 2 \]... y(0) = 2 ...✓ Proved
- \[ \left. \frac{d}{d x} \left(2 - 7 x\right) e^{2 x} \right|_{\substack{ x=0 }} = -3 \]y′(0) = -3.✓ Proved
Answer \( y = \left(2 - 7 x\right) e^{2 x} \)
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) — The solution skips the step of determining the constants C₁ and C₂ from the initial conditions. It merely states that the conditions fix them and then verifies the final answer, rather than deriving it.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-20 — The solution skips the step of determining the constants C₁ and C₂ from the initial conditions. It merely states that the conditions fix them and then verifies the final answer, rather than deriving it.gpt-oss:20b: pass 2026-10-20qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the repeated root, states the general form, and verifies the specific solution against the differential equation and initial conditions.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the repeated root of the characteristic equation, states the general form of the solution, and verifies that the specific answer satisfies both the differential equation and the initial conditions.gpt-oss:20b: pass 2026-10-19
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 2026-10-20 with SymPy 1.14.0.