Home›Calculus 3›Second-order linear equations›Problem 6.205

Second-order linear equations

Problem 6.205 · hard

Solve \( \displaystyle y'' - 4y' + 4y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = -3 \).

Try y = e^(mx): the characteristic equation is m² + bm + c = 0.
Reviewed
\[ \left(m - 2\right)^{2} = m^{2} - 4 m + 4 \]
Roots: 2.✓ Proved
A repeated root r gives y = (C₁ + C₂x)e^(rx).
Reviewed
Use y(0) and y'(0) to find C₁ and C₂.
Reviewed
\[ \left(4 - 20 x\right) e^{2 x} - 4 \frac{d}{d x} \left(1 - 5 x\right) e^{2 x} + \frac{d^{2}}{d x^{2}} \left(1 - 5 x\right) e^{2 x} = 0 \]
The solution satisfies the equation.✓ Proved
\[ 1 \]
y(0).✓ Proved
\[ -3 \]
y'(0).✓ Proved

Answer \( y = \left(1 - 5 x\right) e^{2 x} \)

Lines: 4 proved, 3 reviewed. 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	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	✓ 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: pass — The solution correctly identifies the repeated root, applies the standard form for the general solution, and verifies the initial conditions and differential equation. The steps are logically sound and mathematically correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly identifies the repeated root, applies the standard form for the general solution, and verifies the initial conditions and differential equation. The steps are logically sound and mathematically correct.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution skips the crucial step of determining the constants C1 and C2 using the initial conditions. It jumps from the general form to the specific answer without showing the calculation (C1=1, C2=-5), making the derivation incomplete and logically disconnected.
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.