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Differential equations by direct integration

Problem 6.685 · medium

Find the general solution of \( \displaystyle y'' = 2 x e^{x^{2}} \).

\[ \frac{d}{d x} e^{x^{2}} = 2 x e^{x^{2}} \]
Integrate once: y′ = exp(x**2) + C₁.✓ Proved
\[ \frac{d}{d x} \frac{\sqrt{\pi} \operatorname{erfi}{\left(x \right)}}{2} = e^{x^{2}} \]
Integrate again: y = sqrt(pi)*erfi(x)/2 + C₁x + C₂.✓ Proved

Answer \( y = C_{1} x + C_{2} + \frac{\sqrt{\pi} \operatorname{erfi}{\left(x \right)}}{2} \)

✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. 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
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 antiderivatives and applies the fundamental theorem of calculus twice. The use of the imaginary error function erfi is appropriate for integrating e^{x^2}.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the antiderivatives and applies the fundamental theorem of calculus twice. The use of the imaginary error function erfi is appropriate for integrating e^{x^2}.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-17
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the antiderivatives and applies the constant of integration at each step. The final expression matches the stated answer.
gpt-oss:20b: pass 2026-10-17

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 2026-10-19 with SymPy 1.14.0.