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Differential equations by direct integration

Problem 6.532 · easy

Solve \( \displaystyle y' = 3 x + e^{x} \) with \( \displaystyle y(0) = 3 \).

\[ \frac{d}{d x} \left(\frac{3 x^{2}}{2} + e^{x}\right) = 3 x + e^{x} \]
Integrate: y = this + C.✓ Proved
\[ 3 - \left. \frac{3 x^{2}}{2} + e^{x} \right|_{\substack{ x=0 }} = 2 \]
y(0) = 3 fixes C.✓ Proved
\[ \left. \frac{3 x^{2}}{2} + e^{x} + 2 \right|_{\substack{ x=0 }} = 3 \]
The initial condition holds.✓ Proved

Answer \( y = \frac{3 x^{2}}{2} + e^{x} + 2 \)

✓ 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
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 general antiderivative, determines the constant of integration using the initial condition, and verifies the result.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the general antiderivative, determines the constant of integration using the initial condition, and verifies the result.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the general antiderivative, determines the constant of integration using the initial condition, and verifies the result. The logic is sound.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the general antiderivative, determines the constant of integration using the initial condition, and verifies the final result.
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/de_direct_integration, checked 2026-10-18 with SymPy 1.14.0.