Differential equations by direct integration
Problem 6.427 · easy
Solve \( \displaystyle y' = \frac{1}{x} \) with \( \displaystyle y(1) = 8 \).
- \[ \frac{d}{d x} \ln{\left(x \right)} = \frac{1}{x} \]Integrate: y = this + C.✓ Proved
- \[ 8 - \left. \ln{\left(x \right)} \right|_{\substack{ x=1 }} = 8 \]y(1) = 8 fixes C.✓ Proved
- \[ \left. \ln{\left(x \right)} + 8 \right|_{\substack{ x=1 }} = 8 \]The initial condition holds.✓ Proved
Answer \( y = \ln{\left(x \right)} + 8 \)
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 state the general solution y = ln(x) + C before applying the initial condition, making the jump from the derivative check to the specific constant C=8 logically incomplete. Additionally, the notation 'Subs(log(x), x, 1)' in line 2 is ambiguous and does not clearly show the calculation 8 = ln(1) + C => C = 8.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution fails to explicitly state the general solution y = ln(x) + C before applying the initial condition, making the jump from the derivative check to the specific constant C=8 logically incomplete. Additionally, the notation 'Subs(log(x), x, 1)' in line 2 is ambiguous and does not clearly show the calculation 8 = ln(1) + C => C = 8.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to explicitly state the general solution y = ln(x) + C before applying the initial condition. It jumps from the derivative check to verifying the final answer, omitting the crucial step of solving for the constant C (which is actually 8, not just added arbitrarily). While the final answer is correct, the logical flow presented in the sentences is incomplete and skips the derivation of the constant.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the general solution and applies the initial condition to find the constant. The final answer is correct.gpt-oss:20b: fail (error) 2026-10-14 — The step that determines the constant C is incorrect. It states "8 - Subs(log(x), x, 1) = 8" which merely repeats the initial condition but does not solve for C. The correct deduction is log(1)+C=8 ⇒ C=8. The solution as written would mislead a student into thinking the constant is found by that subtraction.
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 2026-10-17 with SymPy 1.14.0.