Home›Calculus 2›Differential equations by direct integration›Problem 6.571

Differential equations by direct integration

Problem 6.571 · easy

Solve \( \displaystyle y' = 2 x e^{x^{2}} \) with \( \displaystyle y(0) = 8 \).

\[ \frac{d}{d x} e^{x^{2}} = 2 x e^{x^{2}} \]
Integrate: y = this + C.✓ Proved
\[ 8 - \left. e^{x^{2}} \right|_{\substack{ x=0 }} = 7 \]
y(0) = 8 fixes C.✓ Proved
\[ \left. e^{x^{2}} + 7 \right|_{\substack{ x=0 }} = 8 \]
The initial condition holds.✓ Proved

Answer \( y = e^{x^{2}} + 7 \)

✓ 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
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, applies the initial condition to find the constant, and verifies the result. The logic is sound.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the antiderivative, applies the initial condition to find the constant, and verifies the result. The logic is sound.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the antiderivative and applies the initial condition to determine the constant of integration. The logic is sound and the steps are valid.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the antiderivative and applies the initial condition to determine the constant of integration. The steps are logically sound and algebraically correct.
gpt-oss:20b: pass 2026-10-15

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 2026-10-18 with SymPy 1.14.0.