Differential equations by direct integration
Problem 6.575 · easy
Solve \( \displaystyle y' = \frac{1}{x} \) with \( \displaystyle y(1) = 10 \).
- \[ \frac{d}{d x} \ln{\left(x \right)} = \frac{1}{x} \]Integrate: y = this + C.✓ Proved
- \[ 10 - \left. \ln{\left(x \right)} \right|_{\substack{ x=1 }} = 10 \]y(1) = 10 fixes C.✓ Proved
- \[ \left. \ln{\left(x \right)} + 10 \right|_{\substack{ x=1 }} = 10 \]The initial condition holds.✓ Proved
Answer \( y = \ln{\left(x \right)} + 10 \)
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: passqwen3.6:27b-mlx: fail (error) — The solution fails to explicitly solve for the constant of integration C. It jumps from the general form y = ln(x) + C to the final answer y = ln(x) + 10 without showing that C = 9 (since 10 = ln(1) + C implies C = 10). The final answer provided is incorrect; it should be y = ln(x) + 9.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to explicitly solve for the constant of integration C. It jumps from the general form y = ln(x) + C to the final answer y = ln(x) + 10 without showing that C = 9 (since 10 = ln(1) + C implies C = 10). The final answer provided is incorrect; it should be y = ln(x) + 9.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to explicitly determine the constant of integration C. It jumps from the general form y = ln(x) + C to the specific solution y = ln(x) + 10 without showing the step C = 10 - ln(1) = 10. While the final answer is correct, the logical derivation of the constant is missing.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to explicitly determine the constant of integration C. It jumps from the general form y = ln(x) + C to the specific solution y = ln(x) + 10 without showing the step C = 10 - ln(1) = 10. While the final answer is correct, the logical derivation of the constant is missing.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.