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Differential equations by direct integration

Problem 6.740 · easy

Solve \( \displaystyle y' = 2 x e^{x^{2}} \) with \( \displaystyle y(0) = 0 \).

\[ \frac{d}{d x} e^{x^{2}} = 2 x e^{x^{2}} \]
Integrate: y = this + C.✓ Proved
\[ - \left. e^{x^{2}} \right|_{\substack{ x=0 }} + 0 = -1 \]
y(0) = 0 fixes C.✓ Proved
\[ \left. e^{x^{2}} - 1 \right|_{\substack{ x=0 }} = 0 \]
The initial condition holds.✓ Proved

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

Every line of this solution was proved by the computer algebra system SymPy. 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	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ 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 match f, and the initial value holds

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the antiderivative and determines the constant of integration using the initial condition. The logic is sound and the steps are valid.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-26 — The solution correctly identifies the antiderivative and determines the constant of integration using the initial condition. The logic is sound and the steps are valid.
gpt-oss:20b: pass 2026-10-26
qwen3.6:27b-mlx: pass 2026-10-25 — The solution correctly identifies the antiderivative and applies the initial condition to determine the constant of integration. The steps are logically sound and algebraically verified.
gpt-oss:20b: pass 2026-10-25
qwen3.6:27b-mlx: fail (error) 2026-10-25 — The solution fails to explicitly state the general solution y = e^{x^2} + C before applying the initial condition. It jumps from recognizing the derivative to checking the final answer, omitting the crucial step of determining the constant of integration C = -1.
gpt-oss:20b: pass 2026-10-25

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/de_direct_integration, checked 2026-10-26 with SymPy 1.14.0.