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Antiderivatives with initial conditions

Problem 3.381 · medium

Solve the initial-value problem \( \displaystyle f'(x) = 5 x^{3} - \frac{3}{x^{2}} \), \( \displaystyle f(1) = 0 \).
  1. \[ \frac{d}{d x} \left(\frac{5 x^{4}}{4} + \frac{3}{x}\right) = 5 x^{3} - \frac{3}{x^{2}} \]
    An antiderivative; every other differs by a constant C.✓ Proved
  2. \[ - \left. \frac{5 x^{4}}{4} + \frac{3}{x} \right|_{\substack{ x=1 }} = - \frac{17}{4} \]
    f(1) = 0 fixes C.✓ Proved
  3. \[ \frac{5 x^{4}}{4} - \frac{17}{4} + \frac{3}{x} \]
    The solution.✓ Proved
Answer \( f(x) = \frac{5 x^{4}}{4} - \frac{17}{4} + \frac{3}{x} \)

✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.

The full receipt
LineStatusChecked byDetail
1✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
2✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
3✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
answer, a second way✓ Checked independentlysympy 1.14.0 + mpmath 1.3.0the answer differentiated back to the given derivative, and the initial values checked

Reviewers

  • gpt-oss:20b: pass
  • qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The antiderivative of -3/x^2 is 3/x, but the solution incorrectly sets up the constant C by subtracting the value of the antiderivative at x=1 instead of adding it. Specifically, f(x) = F(x) + C, so 0 = F(1) + C implies C = -F(1). The solution calculates -F(1) as -17/4, which is correct for C, but the logic in step 2 is presented confusingly as '-Subs(...) = -17/4' implying C = -17/4 directly from a negative substitution, which is algebraically correct but logically opaque. However, the bigger issue is that the final answer is correct, so let's look closer. F(x) = 5x^4/4 + 3/x. F(1) = 5/4 + 3 = 17/4. C = -17/4. The final function is 5x^4/4 + 3/x - 17/4. The stated answer is 5x^4/4 - 17/4 + 3/x. This is correct. Why fail? Let's re-read the prompt's constraints. 'Check that... each conclusion follows from the lines before it'. Step 2 says '-Subs(...) = -17/4 -- f(1)=0 fixes C'. This implies C = -17/4. Step 3 writes the solution as 5x^4/4 - 17/4 + 3/x. This matches F(x) + C. The algebra is correct. Is there a subtle error? The problem asks to solve the IVP. The solution provides the correct function. The steps are: 1. Find general antiderivative. 2. Determine C. 3. Write solution. The notation in step 2 is weird ('-Subs') but the result is correct. Wait, look at the derivative in step 1. Derivative(5*x**4/4 + 3/x, x) = 5*x**3 - 3/x**2. This is correct. Step 2: -Subs(5*x**4/4 + 3/x, x, 1) = -17/4. This calculates -F(1). Since f(x) = F(x) + C and f(1)=0, C = -F(1). So C = -17/4. This is correct. Step 3 combines them. The solution is actually correct. Why would it fail? Perhaps the 'sentence' part is missing or implied? The prompt says 'Each line... is either an EQUATION... or a SENTENCE'. The text after '--' is the sentence. 'An antiderivative; every other differs by a constant C.' This is a correct statement. 'f(1) = 0 fixes C.' This is a correct statement. 'The solution.' This is a correct statement. Is there a domain issue? x=0 is excluded. The problem doesn't specify domain, but x=1 is in the domain. The solution is valid for x!=0. This is standard. I will pass it unless I find a subtle flaw. Let's look at the sign of the term 3/x. Integral of -3x^-2 is -3 * (x^-1 / -1) = 3x^-1 = 3/x. Correct. F(1) = 5/4 + 3 = 17/4. C = -17/4. Final: 5x^4/4 + 3/x - 17/4. Correct. I see no error.
Every verdict on record (4)
  • qwen3.6:27b-mlx: fail (style) 2026-10-04 — [domain objection, downgraded to style] The antiderivative of -3/x^2 is 3/x, but the solution incorrectly sets up the constant C by subtracting the value of the antiderivative at x=1 instead of adding it. Specifically, f(x) = F(x) + C, so 0 = F(1) + C implies C = -F(1). The solution calculates -F(1) as -17/4, which is correct for C, but the logic in step 2 is presented confusingly as '-Subs(...) = -17/4' implying C = -17/4 directly from a negative substitution, which is algebraically correct but logically opaque. However, the bigger issue is that the final answer is correct, so let's look closer. F(x) = 5x^4/4 + 3/x. F(1) = 5/4 + 3 = 17/4. C = -17/4. The final function is 5x^4/4 + 3/x - 17/4. The stated answer is 5x^4/4 - 17/4 + 3/x. This is correct. Why fail? Let's re-read the prompt's constraints. 'Check that... each conclusion follows from the lines before it'. Step 2 says '-Subs(...) = -17/4 -- f(1)=0 fixes C'. This implies C = -17/4. Step 3 writes the solution as 5x^4/4 - 17/4 + 3/x. This matches F(x) + C. The algebra is correct. Is there a subtle error? The problem asks to solve the IVP. The solution provides the correct function. The steps are: 1. Find general antiderivative. 2. Determine C. 3. Write solution. The notation in step 2 is weird ('-Subs') but the result is correct. Wait, look at the derivative in step 1. Derivative(5*x**4/4 + 3/x, x) = 5*x**3 - 3/x**2. This is correct. Step 2: -Subs(5*x**4/4 + 3/x, x, 1) = -17/4. This calculates -F(1). Since f(x) = F(x) + C and f(1)=0, C = -F(1). So C = -17/4. This is correct. Step 3 combines them. The solution is actually correct. Why would it fail? Perhaps the 'sentence' part is missing or implied? The prompt says 'Each line... is either an EQUATION... or a SENTENCE'. The text after '--' is the sentence. 'An antiderivative; every other differs by a constant C.' This is a correct statement. 'f(1) = 0 fixes C.' This is a correct statement. 'The solution.' This is a correct statement. Is there a domain issue? x=0 is excluded. The problem doesn't specify domain, but x=1 is in the domain. The solution is valid for x!=0. This is standard. I will pass it unless I find a subtle flaw. Let's look at the sign of the term 3/x. Integral of -3x^-2 is -3 * (x^-1 / -1) = 3x^-1 = 3/x. Correct. F(1) = 5/4 + 3 = 17/4. C = -17/4. Final: 5x^4/4 + 3/x - 17/4. Correct. I see no error.
  • gpt-oss:20b: pass 2026-10-04
  • qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the antiderivative and determines the constant of integration using the initial condition. The logic is sound and the final answer is correct.
  • gpt-oss:20b: pass 2026-10-04

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/antiderivative_ivp, checked 2026-10-04 with SymPy 1.14.0.

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