Home›Calculus 3›Second-order linear equations›Problem 6.206

Second-order linear equations

Problem 6.206 · hard

Solve \( \displaystyle y'' + 0y' + 4y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = 2 \).

Try y = e^(mx): the characteristic equation is m² + bm + c = 0.
\[ \left(m - 2 i\right) \left(m + 2 i\right) = m^{2} + 4 \]
Roots: -2*I, 2*I.✓ Proved
Roots α ± βi give y = e^(αx)(C₁cos βx + C₂sin βx).
Use y(0) and y'(0) to find C₁ and C₂.
\[ 4 \sqrt{2} \sin{\left(2 x + \frac{\pi}{4} \right)} + \frac{d^{2}}{d x^{2}} \sqrt{2} \sin{\left(2 x + \frac{\pi}{4} \right)} = 0 \]
The solution satisfies the equation.✓ Proved
\[ 1 \]
y(0).✓ Proved
\[ 2 \]
y'(0).✓ Proved

Answer \( y = \sqrt{2} \sin{\left(2 x + \frac{\pi}{4} \right)} \)

Lines: 4 proved, 3 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.

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	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
7	✓ 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	sympy.checkodesol substitutes the solution back; both initial conditions match

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (misleading) — The solution fails to demonstrate how the constants C1 and C2 were derived from the initial conditions, jumping directly to the final answer. This omits the crucial pedagogical step of solving the system of linear equations, leaving the student without a method to replicate the process for different initial conditions.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (misleading) 2026-10-03 — The solution fails to demonstrate how the constants C1 and C2 were derived from the initial conditions, jumping directly to the final answer. This omits the crucial pedagogical step of solving the system of linear equations, leaving the student without a method to replicate the process for different initial conditions.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution fails to derive the constants C1 and C2 from the initial conditions, instead jumping to the final answer and verifying it. It does not show the work required to solve the problem as stated.
gpt-oss:20b: pass 2026-10-03

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 2026-10-03 with SymPy 1.14.0.