Differential equations by direct integration
Problem 6.680 · easy
Solve \( \displaystyle y' = \frac{1}{x} \) with \( \displaystyle y(1) = 5 \).
- \[ \frac{d}{d x} \ln{\left(x \right)} = \frac{1}{x} \]Integrate: y = this + C.✓ Proved
- \[ 5 - \left. \ln{\left(x \right)} \right|_{\substack{ x=1 }} = 5 \]y(1) = 5 fixes C.✓ Proved
- \[ \left. \ln{\left(x \right)} + 5 \right|_{\substack{ x=1 }} = 5 \]The initial condition holds.✓ Proved
Answer \( y = \ln{\left(x \right)} + 5 \)
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: fail (misleading) — The solution skips the explicit step of solving for the constant C. It merely shows that 5−log(1)=5, but does not derive C=5, which is necessary to justify the final form y=ln(x)+5.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly determine the constant of integration C. It assumes C=5 based on the initial condition but does not show the calculation y(1) = ln(1) + C = 5 => C = 5. The final answer is correct, but the logical step of solving for C is missing.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution fails to explicitly determine the constant of integration C. It assumes C=5 based on the initial condition but does not show the calculation y(1) = ln(1) + C = 5 => C = 5. The final answer is correct, but the logical step of solving for C is missing.gpt-oss:20b: fail (misleading) 2026-10-19 — The solution skips the explicit step of solving for the constant C. It merely shows that 5−log(1)=5, but does not derive C=5, which is necessary to justify the final form y=ln(x)+5.qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the antiderivative and applies the initial condition to determine the constant of integration. The logic is sound and the final answer is correct.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution fails to explicitly solve for the constant of integration C. While the final answer is correct, the steps shown (checking the derivative and verifying the initial condition) do not constitute a derivation of the constant; they merely verify the result. A student would not learn how to find C from these steps.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.