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Second-order linear equations

Problem 6.180 · medium

Solve \( \displaystyle y'' + 0y' - 1y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = 3 \).

Try y = e^(mx): the characteristic equation is m² + bm + c = 0.
Reviewed
\[ \left(m - 1\right) \left(m + 1\right) = m^{2} - 1 \]
Roots: -1, 1.✓ Proved
Two real roots give y = C₁e^(r₁x) + C₂e^(r₂x).
Reviewed
Use y(0) and y'(0) to find C₁ and C₂.
Reviewed
\[ - 3 \sinh{\left(x \right)} - \cosh{\left(x \right)} + \frac{d^{2}}{d x^{2}} \left(3 \sinh{\left(x \right)} + \cosh{\left(x \right)}\right) = 0 \]
The solution satisfies the equation.✓ Proved
\[ 1 \]
y(0).Reviewed
\[ 3 \]
y'(0).Reviewed

Answer \( y = 3 \sinh{\left(x \right)} + \cosh{\left(x \right)} \)

✓ Nihil obstat Lines: 2 proved, 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
4	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
5	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
6	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
7	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the characteristic equation and roots, and the final answer satisfies both the differential equation and initial conditions.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the characteristic equation and roots, and the final answer satisfies both the differential equation and initial conditions.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the characteristic equation, roots, and general form. The verification steps confirm that the stated answer satisfies both the differential equation and the initial conditions.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the characteristic equation, finds the roots, and verifies that the stated answer satisfies the differential equation and initial conditions.
gpt-oss:20b: pass 2026-09-29

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.