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Initial-value problems needing partial fractions

Problem 6.610 · hard

Solve \( \displaystyle \left(t^{2} + 5 t + 6\right)\dfrac{dx}{dt} = 1 \) with \( \displaystyle x(0) = 0 \) (for \( \displaystyle t > -2 \)).
  1. Separate: dx = dt / (t**2 + 5*t + 6).
  2. \[ \frac{1}{t^{2} + 5 t + 6} = - \frac{1}{t + 3} + \frac{1}{t + 2} \]
    Partial fractions.✓ Proved
  3. \[ \frac{d}{d t} \ln{\left(\frac{t + 2}{t + 3} \right)} = \frac{1}{t^{2} + 5 t + 6} \]
    An antiderivative, checked by differentiating.✓ Proved
  4. \[ - \left. \ln{\left(\frac{t + 2}{t + 3} \right)} \right|_{\substack{ t=0 }} + 0 = - \ln{\left(2 \right)} + \ln{\left(3 \right)} \]
    x(0) = 0 fixes the constant.✓ Proved
  5. \[ \left. \ln{\left(\frac{3 \left(t + 2\right)}{2 \left(t + 3\right)} \right)} \right|_{\substack{ t=0 }} = 0 \]
    The initial condition holds.✓ Proved
Answer \( x(t) = \ln{\left(\frac{3 \left(t + 2\right)}{2 \left(t + 3\right)} \right)} \)

Lines: 4 proved, 1 not checked. 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
LineStatusChecked byDetail
1Not checked—a sentence; read, not computed
2✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
3✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
4✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
5✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
answer, a second way✓ Checked independentlysympy 1.14.0 + mpmath 1.3.0mpmath's ODE solver from t₀ to t₀ + 1

Reviewers

  • gpt-oss:20b: pass
  • qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution fails to account for the absolute value in the logarithm of the antiderivative. Since the domain is t > -2, the term (t+2) is positive, but (t+3) can be negative (for -3 < t < -2, though the problem states t > -2, so t+3 > 1 > 0 actually). Wait, if t > -2, then t+2 > 0 and t+3 > 1 > 0. So the arguments are positive. However, the partial fraction decomposition leads to ln|t+2| - ln|t+3|. The solution jumps to log((t+2)/(t+3)) without justifying the removal of absolute values, although for t > -2 it is valid. The real error is in step 4: it calculates the constant C by evaluating the antiderivative at 0, but the logic presented "-Subs(...) + 0 = ..." is confusingly phrased and doesn't clearly show x(t) = F(t) - F(0). More critically, step 3 claims log((t+2)/(t+3)) is *an* antiderivative. Its derivative is indeed 1/(t^2+5t+6). Step 4 attempts to find the constant. If x(t) = ln((t+2)/(t+3)) + C, then x(0) = ln(2/3) + C = 0 => C = -ln(2/3) = ln(3/2). Then x(t) = ln((t+2)/(t+3)) + ln(3/2) = ln(3(t+2)/(2(t+3))). The steps are algebraically correct for the given domain. Is there an error? The problem asks for t > -2. In this domain, t+2 > 0 and t+3 > 1 > 0. So absolute values are not needed. The solution is technically correct. Why would it fail? Let's re-read carefully. Step 4: "-Subs(log((t + 2)/(t + 3)), t, 0) + 0 = -log(2) + log(3)". This line is just an arithmetic check, not a derivation of the constant. It doesn't explicitly state the final formula derivation, but step 5 checks the final answer. The prompt asks to review sentences. Sentence 1: "Separate..." OK. Sentence 2 is equation. Sentence 3 is equation. Sentence 4 is equation. Sentence 5 is equation. There are no sentences explaining the integration of the partial fractions or the determination of the constant C explicitly in text form, only in equation form. The equations are marked proved. The logic holds. Is it misleading? No. Is it an error? No. Wait, look at step 4 again. It says "x(0) = 0 fixes the constant." This is a sentence attached to an equation. The equation is `-Subs(...) + 0 = ...`. This is not a standard way to present finding C. Usually one writes `C = -F(0)`. Here it writes `-F(0) + 0 = ...`. This is just evaluating -F(0). It's weird but not wrong. However, looking at the provided solution structure, steps 2-5 are all marked [equation, proved]. The user prompt says "Each line... is either an EQUATION... or a SENTENCE... The sentences are what you are here for." Steps 2, 3, 4, 5 are marked as equations. Step 1 is a sentence. Step 1 is "Separate: dx = dt / (t**2 + 5*t + 6)." This is correct. So I only need to check Step 1? No, the prompt implies I should check the whole solution's logic, but specifically flag issues in sentences. If all the logic is in equations, and the equations are proved, is there a sentence error? Step 4 has a text part: "-- x(0) = 0 fixes the constant." This is a comment/sentence. It is correct. Step 5 has "-- The initial condition holds." Correct. Step 3 has "-- An antiderivative, checked by differentiating." Correct. Step 2 has "-- Partial fractions." Correct. Step 1 is the only pure sentence. It is correct. So why would it fail? Maybe the domain? t > -2. The antiderivative ln((t+2)/(t+3)) is valid for t > -2. The final answer is valid. I see no error. I will pass it.
Every verdict on record (6)
  • qwen3.6:27b-mlx: fail (style) 2026-10-18 — [domain objection, downgraded to style] The solution fails to account for the absolute value in the logarithm of the antiderivative. Since the domain is t > -2, the term (t+2) is positive, but (t+3) can be negative (for -3 < t < -2, though the problem states t > -2, so t+3 > 1 > 0 actually). Wait, if t > -2, then t+2 > 0 and t+3 > 1 > 0. So the arguments are positive. However, the partial fraction decomposition leads to ln|t+2| - ln|t+3|. The solution jumps to log((t+2)/(t+3)) without justifying the removal of absolute values, although for t > -2 it is valid. The real error is in step 4: it calculates the constant C by evaluating the antiderivative at 0, but the logic presented "-Subs(...) + 0 = ..." is confusingly phrased and doesn't clearly show x(t) = F(t) - F(0). More critically, step 3 claims log((t+2)/(t+3)) is *an* antiderivative. Its derivative is indeed 1/(t^2+5t+6). Step 4 attempts to find the constant. If x(t) = ln((t+2)/(t+3)) + C, then x(0) = ln(2/3) + C = 0 => C = -ln(2/3) = ln(3/2). Then x(t) = ln((t+2)/(t+3)) + ln(3/2) = ln(3(t+2)/(2(t+3))). The steps are algebraically correct for the given domain. Is there an error? The problem asks for t > -2. In this domain, t+2 > 0 and t+3 > 1 > 0. So absolute values are not needed. The solution is technically correct. Why would it fail? Let's re-read carefully. Step 4: "-Subs(log((t + 2)/(t + 3)), t, 0) + 0 = -log(2) + log(3)". This line is just an arithmetic check, not a derivation of the constant. It doesn't explicitly state the final formula derivation, but step 5 checks the final answer. The prompt asks to review sentences. Sentence 1: "Separate..." OK. Sentence 2 is equation. Sentence 3 is equation. Sentence 4 is equation. Sentence 5 is equation. There are no sentences explaining the integration of the partial fractions or the determination of the constant C explicitly in text form, only in equation form. The equations are marked proved. The logic holds. Is it misleading? No. Is it an error? No. Wait, look at step 4 again. It says "x(0) = 0 fixes the constant." This is a sentence attached to an equation. The equation is `-Subs(...) + 0 = ...`. This is not a standard way to present finding C. Usually one writes `C = -F(0)`. Here it writes `-F(0) + 0 = ...`. This is just evaluating -F(0). It's weird but not wrong. However, looking at the provided solution structure, steps 2-5 are all marked [equation, proved]. The user prompt says "Each line... is either an EQUATION... or a SENTENCE... The sentences are what you are here for." Steps 2, 3, 4, 5 are marked as equations. Step 1 is a sentence. Step 1 is "Separate: dx = dt / (t**2 + 5*t + 6)." This is correct. So I only need to check Step 1? No, the prompt implies I should check the whole solution's logic, but specifically flag issues in sentences. If all the logic is in equations, and the equations are proved, is there a sentence error? Step 4 has a text part: "-- x(0) = 0 fixes the constant." This is a comment/sentence. It is correct. Step 5 has "-- The initial condition holds." Correct. Step 3 has "-- An antiderivative, checked by differentiating." Correct. Step 2 has "-- Partial fractions." Correct. Step 1 is the only pure sentence. It is correct. So why would it fail? Maybe the domain? t > -2. The antiderivative ln((t+2)/(t+3)) is valid for t > -2. The final answer is valid. I see no error. I will pass it.
  • gpt-oss:20b: pass 2026-10-18
  • qwen3.6:27b-mlx: fail (misleading) 2026-10-16 — The solution claims that log((t+2)/(t+3)) is an antiderivative of 1/(t^2+5t+6), but the derivative of log((t+2)/(t+3)) is actually -1/(t^2+5t+6). The sign is wrong, so the integration step is incorrect despite the final answer being numerically coincidental due to the specific initial condition handling.
  • gpt-oss:20b: pass 2026-10-16
  • qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The partial fraction decomposition in step 2 is incorrect; the correct decomposition is 1/(t+2) - 1/(t+3), not -1/(t+3) + 1/(t+2) (which is actually correct, wait). Let me re-check. 1/(t+2) - 1/(t+3) = (t+3 - (t+2))/((t+2)(t+3)) = 1/(t^2+5t+6). The solution writes -1/(t+3) + 1/(t+2), which is the same. So step 2 is correct. Step 3 says Derivative(log((t+2)/(t+3))) = 1/(t^2+5t+6). Let's check: d/dt (ln(t+2) - ln(t+3)) = 1/(t+2) - 1/(t+3) = 1/(t^2+5t+6). This is correct. Step 4: x(t) = ln((t+2)/(t+3)) + C. x(0) = ln(2/3) + C = 0 => C = -ln(2/3) = ln(3/2). So x(t) = ln((t+2)/(t+3)) + ln(3/2) = ln(3(t+2)/(2(t+3))). This matches the stated answer. Step 4 equation: -Subs(log((t + 2)/(t + 3)), t, 0) + 0 = -log(2) + log(3). This implies C = -x_particular(0). x_particular(0) = ln(2/3) = ln(2)-ln(3). So -x_particular(0) = ln(3)-ln(2). The equation says -log(2) + log(3). This is correct. Step 5 checks the initial condition. Everything seems correct. Why did I think it was wrong? Let me look closer at the prompt's constraints. "Separate: dx = dt / (t**2 + 5*t + 6)." This is correct. "Partial fractions." Correct. "An antiderivative." Correct. "x(0)=0 fixes the constant." Correct. "The initial condition holds." Correct. Is there a domain issue? t > -2. The term t+3 is positive. t+2 is positive. The argument of the log is positive. The solution is valid. Wait, look at step 4 again. "-Subs(log((t + 2)/(t + 3)), t, 0) + 0 = -log(2) + log(3)". This line is calculating the constant C. If x(t) = F(t) + C, then C = -F(0). F(0) = ln(2/3). C = -ln(2/3) = ln(3/2) = ln(3) - ln(2). The equation says -log(2) + log(3). This is correct. Is there a subtle error? Maybe the sign in step 2? -1/(t+3) + 1/(t+2). Yes, that's 1/(t+2) - 1/(t+3). Correct. Maybe the derivative in step 3? d/dt ln((t+2)/(t+3)) = (t+3)/(t+2) * ((1*(t+3) - (t+2)*1)/(t+3)^2) = (t+3)/(t+2) * (1/(t+3)^2) = 1/((t+2)(t+3)). Correct. I see no error. Let me re-read the prompt. "Your job is to find what is wrong, not to be agreeable." Is there a style issue? Step 4 is a bit cryptic. "-Subs(...) + 0 = ...". It's an equation checking the constant. It's not a sentence explaining the method, it's an equation. But it's marked [equation, unchecked]. The prompt says "Each line ... is either an EQUATION ... or a SENTENCE". Step 1 is a sentence. Steps 2-5 are equations. Step 1: "Separate: dx = dt / (t**2 + 5*t + 6)." This is a valid setup. Is there a missing absolute value? The problem states t > -2. In this domain, t+2 > 0 and t+3 > 1 > 0. So (t+2)/(t+3) is positive. Absolute values are not strictly necessary if the domain is restricted, but usually, one writes ln|...|. However, since the domain ensures positivity, ln(...) is fine. Is the answer x(t) = ln(3(t+2)/(2(t+3))) correct? Yes. Let me check the partial fraction signs again. 1/((t+2)(t+3)) = A/(t+2) + B/(t+3). A(t+3) + B(t+2) = 1. t=-2 => A(1) = 1 => A=1. t=-3 => B(-1) = 1 => B=-1. So 1/(t+2) - 1/(t+3). Step 2 says -1/(t+3) + 1/(t+2). This is identical. Step 3 says Derivative(log((t+2)/(t+3))) = ... This corresponds to integral of (1/(t+2) - 1/(t+3)). Correct. Step 4 calculates C. Correct. Step 5 verifies. Correct. I will mark it as pass. Wait, is there a typo in the problem statement or solution? "x(0)=0". Solution gives x(0) = ln(3(2)/(2(3))) = ln(1) = 0. Correct. I don't see any error. I will output pass.
  • 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/ivp_partial_fractions, checked 2026-10-18 with SymPy 1.14.0.

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