Home›Calculus 3›Second-order initial and boundary value problems›Problem 6.477

Second-order initial and boundary value problems

Problem 6.477 · medium

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

\[ m^{2} + 3 m - 4 = \left(m - 1\right) \left(m + 4\right) \]
The characteristic equation has distinct real roots -4 and 1.✓ Proved
So y = C₁exp(-4*x) + C₂exp(x); the two conditions fix C₁ and C₂.
\[ - 16 e^{x} + 3 \frac{d}{d x} 4 e^{x} + \frac{d^{2}}{d x^{2}} 4 e^{x} = 0 \]
It satisfies the equation ...✓ Proved
\[ \left. 4 e^{x} \right|_{\substack{ x=0 }} = 4 \]
... y(0) = 4 ...✓ Proved
\[ \left. \frac{d}{d x} 4 e^{x} \right|_{\substack{ x=0 }} = 4 \]
y′(0) = 4.✓ Proved

Answer \( y = 4 e^{x} \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to determine the constants C₁ and C₂ from the initial conditions. It merely verifies that the specific function y = 4e^x satisfies the conditions, but does not show that this is the unique solution derived from the general form y = C₁e^{-4x} + C₂e^x. A student might incorrectly conclude that one can simply guess a solution and verify it, rather than solving the system of linear equations for the constants.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution fails to determine the constants C₁ and C₂ from the initial conditions. It merely verifies that the specific function y = 4e^x satisfies the conditions, but does not show that this is the unique solution derived from the general form y = C₁e^{-4x} + C₂e^x. A student might incorrectly conclude that one can simply guess a solution and verify it, rather than solving the system of linear equations for the constants.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to determine the constants C₁ and C₂ from the initial conditions. It merely verifies that the final answer satisfies the conditions, ignoring the general solution structure derived in step 2. Specifically, it skips the system of equations C₁ + C₂ = 4 and -4C₁ + C₂ = 4, which yields C₁ = 0 and C₂ = 4. While the final answer is correct, the logical gap implies the student did not actually solve for the constants.
gpt-oss:20b: fail (error) 2026-10-15 — The solution skips the crucial step of determining the constants C₁ and C₂ from the initial conditions. It merely states that the conditions fix them but never actually solves for C₁=0 and C₂=4, and then proceeds to test only the particular function y=4e^x without showing that it is the unique solution that satisfies both the differential equation and the given initial values.
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to determine the constants C₁ and C₂ from the initial conditions. It merely verifies that the specific function y = 4e^x is a solution, but does not show that it is the unique solution satisfying the given initial conditions (specifically, it skips the step showing C₁ must be 0).
gpt-oss:20b: pass 2026-10-14

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 2026-10-17 with SymPy 1.14.0.