Home›Calculus 3›Second-order linear equations›Problem 6.207

Second-order linear equations

Problem 6.207 · hard

Solve \( \displaystyle y'' - 1y' - 2y = 0 \) with \( \displaystyle y(0) = -3,\ y'(0) = -2 \).

Try y = e^(mx): the characteristic equation is m² + bm + c = 0.
\[ \left(m - 2\right) \left(m + 1\right) = m^{2} - m - 2 \]
Roots: 2, -1.✓ Proved
Two real roots give y = C₁e^(r₁x) + C₂e^(r₂x).
Use y(0) and y'(0) to find C₁ and C₂.
\[ \frac{\left(10 e^{3 x} + 8\right) e^{- x}}{3} - \frac{d}{d x} \frac{\left(- 5 e^{3 x} - 4\right) e^{- x}}{3} + \frac{d^{2}}{d x^{2}} \frac{\left(- 5 e^{3 x} - 4\right) e^{- x}}{3} = 0 \]
The solution satisfies the equation.✓ Proved
\[ -3 \]
y(0).✓ Proved
\[ -2 \]
y'(0).✓ Proved

Answer \( y = \frac{\left(- 5 e^{3 x} - 4\right) e^{- x}}{3} \)

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 (error) — The solution fails to actually solve for the constants C1 and C2; it merely verifies that the final answer satisfies the ODE and initial conditions. A student would not learn how to determine the constants from the initial values.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution fails to actually solve for the constants C1 and C2; it merely verifies that the final answer satisfies the ODE and initial conditions. A student would not learn how to determine the constants from the initial values.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution fails to actually solve for the constants C1 and C2. It states the method (line 4) but skips the algebraic steps, jumping directly to verification equations (lines 5-7) that assume the final answer is correct without deriving it.
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.