Initial-value problems needing partial fractions

Reviewers

• gpt-oss:20b: pass
• qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution fails to account for the absolute values in the logarithmic integration, which is critical given the domain t > 5. While the final expression happens to be correct for t > 5, the intermediate step (line 3) presents an antiderivative that is undefined or complex for 5 < t < 2 (though the domain is t>5, the general form log((t-5)/(t+2)) is negative, and standard integration yields log|...|). More importantly, line 4 calculates the constant C incorrectly based on the form in line 3. Line 3 gives F(t) = (1/7)ln((t-5)/(t+2)). At t=6, F(6) = (1/7)ln(1/8) = -(ln 8)/7. To satisfy x(6)=0, we need x(t) = F(t) - F(6) = (1/7)ln((t-5)/(t+2)) + (ln 8)/7 = (1/7)ln(8(t-5)/(t+2)). Line 4 claims '-Subs(...) = 3*log(2)/7', which is numerically correct for the constant term, but the logic is opaque and relies on the specific evaluation rather than showing the general constant determination. However, the primary error is that line 3 states an equality of derivatives that holds, but the integration step from 1 to 3 skips the absolute value, which is a standard requirement for rigorous calculus solutions, especially when the argument of the log can be negative or when defining the domain. Actually, for t>5, (t-5)/(t+2) is positive, so the absolute value isn't strictly necessary for correctness in this specific domain, but the transition from 2 to 3 is presented as a 'checked' fact without showing the integration constant handling clearly in the sentences. Let's look closer at line 4. It says '-Subs(log((t - 5)/(t + 2))/7, t, 6) + 0 = 3*log(2)/7'. This implies C = 3*log(2)/7. Then x(t) = log((t-5)/(t+2))/7 + 3*log(2)/7. This simplifies to log(8(t-5)/(t+2))/7. The math is correct. Is there a subtle error? The problem asks for t > 5. The term t-5 is positive. t+2 is positive. So (t-5)/(t+2) is positive. The log is real. The solution is technically correct. Why would it fail? Perhaps the 'sentence' in line 1 'Separate: dx = dt / ...' is not a sentence but an equation? No, it's marked sentence. Is the partial fraction correct? 1/((t-5)(t+2)) = A/(t-5) + B/(t+2). A(t+2) + B(t-5) = 1. t=5 -> 7A=1 -> A=1/7. t=-2 -> -7B=1 -> B=-1/7. So 1/(7(t-5)) - 1/(7(t+2)). Line 2 says -1/(7(t+2)) + 1/(7(t-5)). This is correct. Line 3 derivative check is correct. Line 4 calculation: log((6-5)/(6+2))/7 = log(1/8)/7 = -log(8)/7 = -3log(2)/7. So -Subs(...) = 3log(2)/7. Correct. Line 5 checks the final answer. The solution seems sound. Wait, look at line 4 again. It says '-Subs(...) + 0 = ...'. This is an equation line marked proved. The prompt says 'Each line ... is either an EQUATION ... or a SENTENCE'. Line 4 is marked equation. Line 1 is marked sentence. Line 1 says 'Separate: dx = dt / ...'. This is a valid setup. Is there a domain issue? t>5. The solution is valid. Maybe the 'error' is that the student didn't explicitly state the absolute value? In many calculus contexts, omitting absolute values when the argument is known to be positive in the domain is acceptable, but often considered 'style' or 'misleading' if not justified. However, the prompt asks for 'error' if a step is wrong. The step is not wrong for t>5. Let's look for a more subtle error. The problem is (t^2 - 3t - 10) dx/dt = 1. dx/dt = 1/(t^2-3t-10). The solution integrates this. The result is x(t) = 1/7 ln((t-5)/(t+2)) + C. x(6) = 0 => 0 = 1/7 ln(1/8) + C => C = -1/7 ln(1/8) = 1/7 ln(8). x(t) = 1/7 ln((t-5)/(t+2)) + 1/7 ln(8) = 1/7 ln(8(t-5)/(t+2)). This matches the stated answer. The solution is correct. Why would I fail it? Perhaps the 'sentence' in line 1 is too brief? Or maybe the fact that line 3 is an 'equation, proved' means I shouldn't check the algebra, but I should check if the sentence leading to it is valid? There is no sentence leading to line 3. Line 2 is an equation. Line 3 is an equation. The jump from 2 to 3 is implicit integration. Usually, a sentence like 'Integrating both sides yields...' is expected. Without it, the logical flow is missing, but is it an 'error'? The prompt says 'Check that ... each conclusion follows from the lines before it'. Line 3 does not follow from Line 2 without the implicit step of integration. This is a gap in reasoning. However, often 'equation, proved' lines are allowed to skip trivial steps. But here, the constant of integration is handled in line 4. The structure is: 1. Separate. 2. PF. 3. Antiderivative found. 4. Constant found. 5. Check. This is a standard workflow. I will pass it unless there is a specific technicality. Wait, look at the domain t>5. The term t-5 is positive. The term t+2 is positive. The ratio is positive. The log is defined. The solution is correct. I will pass.

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 2026-10-30 with SymPy 1.14.0.