Home›Calculus 2›Initial-value problems needing partial fractions›Problem 6.697

Initial-value problems needing partial fractions

Problem 6.697 · hard

Solve \( \displaystyle \left(t^{2} - t - 2\right)\dfrac{dx}{dt} = 1 \) with \( \displaystyle x(3) = 0 \) (for \( \displaystyle t > 2 \)).

Separate: dx = dt / (t**2 - t - 2).
Reviewed
\[ \frac{1}{t^{2} - t - 2} = - \frac{1}{3 t + 3} + \frac{1}{3 t - 6} \]
Partial fractions.✓ Proved
\[ \frac{d}{d t} \frac{\ln{\left(\frac{t - 2}{t + 1} \right)}}{3} = \frac{1}{t^{2} - t - 2} \]
An antiderivative, checked by differentiating.✓ Proved
\[ - \left. \frac{\ln{\left(\frac{t - 2}{t + 1} \right)}}{3} \right|_{\substack{ t=3 }} + 0 = \frac{2 \ln{\left(2 \right)}}{3} \]
x(3) = 0 fixes the constant.✓ Proved
\[ \left. \frac{\ln{\left(\frac{4 \left(t - 2\right)}{t + 1} \right)}}{3} \right|_{\substack{ t=3 }} = 0 \]
The initial condition holds.✓ Proved

Answer \( x(t) = \frac{\ln{\left(\frac{4 \left(t - 2\right)}{t + 1} \right)}}{3} \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
5	✓ 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	mpmath's ODE solver from t₀ to t₀ + 1

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly separates variables, applies partial fraction decomposition, and determines the constant of integration to satisfy the initial condition. The final expression is algebraically equivalent to the stated answer.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly separates variables, applies partial fraction decomposition, and determines the constant of integration to satisfy the initial condition. The final expression is algebraically equivalent to the stated answer.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (style) 2026-10-17 — [domain objection, downgraded to style] The solution fails to account for the absolute value in the logarithm arising from the integration of 1/t. Since the domain is t > 2, t-2 > 0 and t+1 > 0, so |(t-2)/(t+1)| = (t-2)/(t+1). However, the constant determination in step 4 is algebraically inconsistent with the form in step 3 unless the absolute value logic is explicitly handled or the constant is derived correctly from the specific antiderivative chosen. Step 4 calculates the constant C such that x(t) = (1/3)ln((t-2)/(t+1)) + C. At t=3, x(3) = (1/3)ln(2/4) + C = (1/3)ln(1/2) + C = 0 => C = -(1/3)ln(1/2) = (1/3)ln(2). Thus x(t) = (1/3)ln((t-2)/(t+1)) + (1/3)ln(2) = (1/3)ln(2(t-2)/(t+1)). The stated answer has a 4 inside the log, which is incorrect. The check in step 5 verifies the *stated answer* satisfies the IC, but the derivation in steps 3-4 does not lead to that answer; it leads to a different one. Specifically, step 4 claims the constant is 2*log(2)/3, but if we use the antiderivative in step 3, the constant should be log(2)/3. The jump from step 4 to the final answer is not justified by the previous lines.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (style) 2026-10-17 — [domain objection, downgraded to style] The integration constant is incorrect. The general solution is x(t) = (1/3)ln((t-2)/(t+1)) + C. Applying x(3)=0 gives C = -(1/3)ln(1/4) = (1/3)ln(4). The final answer should be x(t) = (1/3)ln(4(t-2)/(t+1)), but the derivation in step 4 incorrectly calculates the constant term as 2*log(2)/3 added to the log term without properly combining it into the single logarithmic expression shown in step 5, or rather, step 4 claims the constant is 2*log(2)/3 but step 5 presents a formula that implies the constant was handled differently (specifically, ln(4) = 2ln(2), so the constant is correct, but the logic in step 4 is confusingly stated as '-Subs... + 0 = ...' which doesn't clearly derive the constant C. Actually, looking closer: Step 4 says '-Subs(log((t - 2)/(t + 1))/3, t, 3) + 0 = 2*log(2)/3'. LHS: -1/3 * ln(1/4) = 1/3 * ln(4) = 2/3 ln(2). So C = 2/3 ln(2). The final answer is x(t) = 1/3 ln((t-2)/(t+1)) + 2/3 ln(2) = 1/3 (ln((t-2)/(t+1)) + ln(4)) = 1/3 ln(4(t-2)/(t+1)). The math is actually correct. Wait, let me re-read the prompt's constraint. 'unchecked' equations are not to be re-checked. I must check the sentences. Sentence 1 is 'Separate: dx = dt / (t**2 - t - 2)'. This is correct. There are no other sentences. The equations are unchecked. If the equations are unchecked, I assume they are correct? No, the prompt says 'Each line... is either an EQUATION... or a SENTENCE... The sentences are what you are here for.' It implies I should check the sentences. Are there any sentences that are wrong? Sentence 1 is correct. The rest are equations. If the equations are unchecked, do I assume they are true? The prompt says 'a computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)'. But these are marked 'unchecked'. This is a contradiction in the prompt instructions vs the data. Usually, 'unchecked' means I should check it. But the instructions say 'do not re-check the algebra of a line marked proved'. These are not marked proved. They are marked 'unchecked'. This likely means I SHOULD check them. Let's check the math. Step 2: Partial fractions. 1/((t-2)(t+1)) = A/(t-2) + B/(t+1). A(t+1) + B(t-2) = 1. t=2 => 3A=1 => A=1/3. t=-1 => -3B=1 => B=-1/3. So 1/(3(t-2)) - 1/(3(t+1)). Step 2 says -1/(3*(t + 1)) + 1/(3*(t - 2)). This is correct. Step 3: Derivative of 1/3 ln((t-2)/(t+1)). d/dt [1/3 (ln(t-2) - ln(t+1))] = 1/3 (1/(t-2) - 1/(t+1)) = 1/3 ((t+1 - (t-2))/(t^2-t-2)) = 1/3 (3/(t^2-t-2)) = 1/(t^2-t-2). Correct. Step 4: x(3)=0. x(t) = 1/3 ln((t-2)/(t+1)) + C. 0 = 1/3 ln(1/4) + C => C = -1/3 ln(1/4) = 1/3 ln(4) = 2/3 ln(2). The equation says '-Subs(log((t - 2)/(t + 1))/3, t, 3) + 0 = 2*log(2)/3'. LHS: -1/3 ln(1/4) = 2/3 ln(2). RHS: 2/3 ln(2). Correct. Step 5: Subs(log(4*(t - 2)/(t + 1))/3, t, 3) = 0. LHS: 1/3 ln(4(1)/4) = 1/3 ln(1) = 0. Correct. The solution is correct. Why would it fail? Maybe the domain? t > 2. t-2 > 0, t+1 > 0. Argument of log is positive. Correct. Is there a sentence error? Sentence 1: 'Separate: dx = dt / (t**2 - t - 2).' This is a valid separation of variables. It seems correct. Wait, is the prompt tricking me with 'unchecked'? If I am not supposed to check unchecked equations, and there are no sentences to check (or only one correct sentence), it should pass. However, often 'unchecked' in these benchmarks implies 'you must check this'. If I must check, and it's correct, it passes. Let's look for subtle errors. 'Separate: dx = dt / ...'. Technically dx/dt = ... so dx = ... dt. The syntax 'dx = dt / ...' is standard shorthand. Is there a sign error? No. Is the constant of integration handled correctly? Yes. I will mark it as pass.
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/ivp_partial_fractions, checked 2026-10-19 with SymPy 1.14.0.