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Differential equations by direct integration

Problem 6.577 · medium

Find the general solution of \( \displaystyle y'' = 3 x + e^{x} \).

\[ \frac{d}{d x} \left(\frac{3 x^{2}}{2} + e^{x}\right) = 3 x + e^{x} \]
Integrate once: y′ = 3*x**2/2 + exp(x) + C₁.✓ Proved
\[ \frac{d}{d x} \left(\frac{x^{3}}{2} + e^{x}\right) = \frac{3 x^{2}}{2} + e^{x} \]
Integrate again: y = x**3/2 + exp(x) + C₁x + C₂.✓ Proved

Answer \( y = C_{1} x + C_{2} + \frac{x^{3}}{2} + e^{x} \)

✓ 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 performs two successive integrations to find the general solution, properly accounting for the constants of integration.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly performs two successive integrations to find the general solution, properly accounting for the constants of integration.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly performs two successive integrations to find the general solution, properly accounting for the constants of integration at each step.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly performs two successive integrations to find the general solution, and the final result matches the stated answer.
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.