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Differential equations by direct integration

Problem 6.739 · easy

Solve \( \displaystyle y' = \frac{1}{x} \) with \( \displaystyle y(1) = 4 \).

\[ \frac{d}{d x} \ln{\left(x \right)} = \frac{1}{x} \]
Integrate: y = this + C.✓ Proved
\[ 4 - \left. \ln{\left(x \right)} \right|_{\substack{ x=1 }} = 4 \]
y(1) = 4 fixes C.✓ Proved
\[ \left. \ln{\left(x \right)} + 4 \right|_{\substack{ x=1 }} = 4 \]
The initial condition holds.✓ Proved

Answer \( y = \ln{\left(x \right)} + 4 \)

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: pass
qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution incorrectly determines the constant of integration. Since ln(1) = 0, the condition y(1) = 4 implies C = 4, making the solution y = ln(x) + 4. However, the logic in step 2 is presented as '4 - Subs(log(x), x, 1) = 4', which is numerically true (4-0=4) but conceptually confusingly phrased as if it were a derivation step rather than a check. More critically, the final answer y = ln(x) + 4 is actually correct, but the explanation in step 2 is misleadingly structured. Wait, let me re-read. y' = 1/x => y = ln|x| + C. y(1)=4 => 4 = ln(1) + C => C=4. So y = ln(x) + 4. The stated answer is correct. The steps are: 1. Integrate to get form. 2. Solve for C. 3. Verify. Step 2 says '4 - Subs(log(x), x, 1) = 4'. This calculates C = 4 - ln(1) = 4. This is correct. Step 3 verifies. The solution is sound. Why did I think it was wrong? Let me look closer. Is there a domain issue? x>0 is implied by ln(x). The problem doesn't specify domain, but ln(x) is standard. Is the notation 'log' natural log? In calculus contexts, usually yes. If it were base 10, it would be wrong. Standard assumption is natural log. The steps are logically sound. Verdict: pass.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-26 — [domain objection, downgraded to style] The solution incorrectly determines the constant of integration. Since ln(1) = 0, the condition y(1) = 4 implies C = 4, making the solution y = ln(x) + 4. However, the logic in step 2 is presented as '4 - Subs(log(x), x, 1) = 4', which is numerically true (4-0=4) but conceptually confusingly phrased as if it were a derivation step rather than a check. More critically, the final answer y = ln(x) + 4 is actually correct, but the explanation in step 2 is misleadingly structured. Wait, let me re-read. y' = 1/x => y = ln|x| + C. y(1)=4 => 4 = ln(1) + C => C=4. So y = ln(x) + 4. The stated answer is correct. The steps are: 1. Integrate to get form. 2. Solve for C. 3. Verify. Step 2 says '4 - Subs(log(x), x, 1) = 4'. This calculates C = 4 - ln(1) = 4. This is correct. Step 3 verifies. The solution is sound. Why did I think it was wrong? Let me look closer. Is there a domain issue? x>0 is implied by ln(x). The problem doesn't specify domain, but ln(x) is standard. Is the notation 'log' natural log? In calculus contexts, usually yes. If it were base 10, it would be wrong. Standard assumption is natural log. The steps are logically sound. Verdict: pass.
gpt-oss:20b: pass 2026-10-26
qwen3.6:27b-mlx: pass 2026-10-25 — 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: fail (misleading) 2026-10-25 — The step claiming "4 - Subs(log(x), x, 1) = 4" does not correctly determine the constant C; it merely shows that 4-0=4, but does not use the initial condition to solve for C. The correct reasoning is y(1)=log(1)+C=4, hence C=4.
qwen3.6:27b-mlx: fail (error) 2026-10-25 — The solution fails to explicitly solve for the constant of integration C. While the final answer is correct, the logic presented (checking that the final answer satisfies the condition) does not demonstrate the derivation of C from the general solution y = ln(x) + C. A student would not learn how to find C.
gpt-oss:20b: pass 2026-10-25

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 2026-10-26 with SymPy 1.14.0.