Home›Calculus 3›Second-order initial and boundary value problems›Problem 6.630

Second-order initial and boundary value problems

Problem 6.630 · medium

Solve \( \displaystyle - 2 y - y' + y'' = 0 \) with \( \displaystyle y(0) = 4,\ y'(0) = -3 \).

\[ m^{2} - m - 2 = \left(m - 2\right) \left(m + 1\right) \]
The characteristic equation has distinct real roots 2 and -1.✓ Proved
So y = C₁exp(2*x) + C₂exp(-x); the two conditions fix C₁ and C₂.
\[ \frac{\left(- 2 e^{3 x} - 22\right) e^{- x}}{3} - \frac{d}{d x} \frac{\left(e^{3 x} + 11\right) e^{- x}}{3} + \frac{d^{2}}{d x^{2}} \frac{\left(e^{3 x} + 11\right) e^{- x}}{3} = 0 \]
It satisfies the equation ...✓ Proved
\[ \left. \frac{\left(e^{3 x} + 11\right) e^{- x}}{3} \right|_{\substack{ x=0 }} = 4 \]
... y(0) = 4 ...✓ Proved
\[ \left. \frac{d}{d x} \frac{\left(e^{3 x} + 11\right) e^{- x}}{3} \right|_{\substack{ x=0 }} = -3 \]
y′(0) = -3.✓ Proved

Answer \( y = \frac{\left(e^{3 x} + 11\right) e^{- x}}{3} \)

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

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Not checked	—	a sentence; read, not computed
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	finite differences of the answer satisfy the equation, and the conditions hold in floating point

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The characteristic equation derived in step 1 is incorrect. For y'' - y' - 2y = 0, the characteristic equation is m^2 - m - 2 = 0, which factors as (m-2)(m+1)=0, giving roots 2 and -1. However, the stated answer y = (e^{3x} + 11)e^{-x}/3 simplifies to (1/3)e^{2x} + (11/3)e^{-x}, which corresponds to roots 2 and -1. Wait, let me re-read carefully. The problem is -2y - y' + y'' = 0, which is y'' - y' - 2y = 0. Roots are 2, -1. General solution C1 e^{2x} + C2 e^{-x}. The stated answer is (e^{3x} + 11)e^{-x}/3 = (1/3)e^{2x} + (11/3)e^{-x}. Let's check ICs. y(0) = 1/3 + 11/3 = 4. Correct. y'(x) = (2/3)e^{2x} - (11/3)e^{-x}. y'(0) = 2/3 - 11/3 = -9/3 = -3. Correct. So the stated answer is actually correct. Why did I think it was wrong? Let me re-read step 1. Step 1 says m^2 - m - 2 = (m-2)(m+1). This is correct. Step 2 says y = C1 exp(2x) + C2 exp(-x). This is correct. Step 3, 4, 5 verify the specific solution. The solution seems correct. Is there a subtle error? The problem asks to 'Solve'. The solution jumps to verifying a specific answer without showing how C1 and C2 were found. Step 2 says 'the two conditions fix C1 and C2' but doesn't show the work. Then steps 3-5 just verify the final answer. This is a valid verification method, but is it 'solving'? Usually, 'solve' implies deriving the constants. However, the prompt asks to review the solution. If the solution is just a verification, is it wrong? The prompt says 'Solve ... Stated answer ...'. The solution provided is a verification of the stated answer. It does not actually solve for C1 and C2. It asserts the form, then verifies the specific instance. This is logically incomplete as a 'solution' derivation, but the steps themselves are correct. However, looking closely at Step 1: 'm**2 - m - 2 = (m - 2)*(m + 1)'. This is an identity. The text says 'The characteristic equation has distinct real roots 2 and -1.' This is correct. Step 2: 'So y = C1exp(2*x) + C2exp(-x)'. Correct. Step 3-5: Verification. The solution fails to actually determine C1 and C2. It just checks the answer. Is this an error? In an adversarial review, skipping the derivation of constants when asked to 'solve' is a significant omission, potentially misleading a student into thinking verification is sufficient or that the constants appear by magic. But is it 'wrong'? The steps shown are correct. The conclusion follows from the verification. But the 'solution' doesn't show how the answer was obtained. Let's look for a harder error. Did I misread the ODE? -2y - y' + y'' = 0 => y'' - y' - 2y = 0. Characteristic eq: r^2 - r - 2 = 0. Roots: (1 +/- sqrt(1+8))/2 = (1 +/- 3)/2 => 2, -1. Correct. The stated answer is y = (e^{3x} + 11)e^{-x}/3 = (1/3)e^{2x} + (11/3)e^{-x}. This matches the form. The ICs match. The verification steps are algebraically correct (marked proved). The only issue is the lack of derivation for C1, C2. Is this 'misleading'? A student might think they don't need to solve for constants. Or is it just 'style'? No, it's a missing step in the logical flow of 'solving'. However, often in these datasets, if the verification is complete, it's considered a valid 'check'. But the prompt says 'Solve'. If I submit a blank paper with just the answer and 'it works', I get no credit. The solution presented is essentially 'Here is the answer, and here is proof it works.' It does not show the work to get the answer. This is a 'fail' because it doesn't actually solve the problem, it only verifies the result. The sentence 'the two conditions fix C1 and C2' is a claim, not a derivation. The subsequent steps do not derive C1 and C2; they verify the final expression. Therefore, the solution is incomplete/misleading as a tutorial.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The characteristic equation derived in step 1 is incorrect. For y'' - y' - 2y = 0, the characteristic equation is m^2 - m - 2 = 0, which factors as (m-2)(m+1)=0, giving roots 2 and -1. However, the stated answer y = (e^{3x} + 11)e^{-x}/3 simplifies to (1/3)e^{2x} + (11/3)e^{-x}, which corresponds to roots 2 and -1. Wait, let me re-read carefully. The problem is -2y - y' + y'' = 0, which is y'' - y' - 2y = 0. Roots are 2, -1. General solution C1 e^{2x} + C2 e^{-x}. The stated answer is (e^{3x} + 11)e^{-x}/3 = (1/3)e^{2x} + (11/3)e^{-x}. Let's check ICs. y(0) = 1/3 + 11/3 = 4. Correct. y'(x) = (2/3)e^{2x} - (11/3)e^{-x}. y'(0) = 2/3 - 11/3 = -9/3 = -3. Correct. So the stated answer is actually correct. Why did I think it was wrong? Let me re-read step 1. Step 1 says m^2 - m - 2 = (m-2)(m+1). This is correct. Step 2 says y = C1 exp(2x) + C2 exp(-x). This is correct. Step 3, 4, 5 verify the specific solution. The solution seems correct. Is there a subtle error? The problem asks to 'Solve'. The solution jumps to verifying a specific answer without showing how C1 and C2 were found. Step 2 says 'the two conditions fix C1 and C2' but doesn't show the work. Then steps 3-5 just verify the final answer. This is a valid verification method, but is it 'solving'? Usually, 'solve' implies deriving the constants. However, the prompt asks to review the solution. If the solution is just a verification, is it wrong? The prompt says 'Solve ... Stated answer ...'. The solution provided is a verification of the stated answer. It does not actually solve for C1 and C2. It asserts the form, then verifies the specific instance. This is logically incomplete as a 'solution' derivation, but the steps themselves are correct. However, looking closely at Step 1: 'm**2 - m - 2 = (m - 2)*(m + 1)'. This is an identity. The text says 'The characteristic equation has distinct real roots 2 and -1.' This is correct. Step 2: 'So y = C1exp(2*x) + C2exp(-x)'. Correct. Step 3-5: Verification. The solution fails to actually determine C1 and C2. It just checks the answer. Is this an error? In an adversarial review, skipping the derivation of constants when asked to 'solve' is a significant omission, potentially misleading a student into thinking verification is sufficient or that the constants appear by magic. But is it 'wrong'? The steps shown are correct. The conclusion follows from the verification. But the 'solution' doesn't show how the answer was obtained. Let's look for a harder error. Did I misread the ODE? -2y - y' + y'' = 0 => y'' - y' - 2y = 0. Characteristic eq: r^2 - r - 2 = 0. Roots: (1 +/- sqrt(1+8))/2 = (1 +/- 3)/2 => 2, -1. Correct. The stated answer is y = (e^{3x} + 11)e^{-x}/3 = (1/3)e^{2x} + (11/3)e^{-x}. This matches the form. The ICs match. The verification steps are algebraically correct (marked proved). The only issue is the lack of derivation for C1, C2. Is this 'misleading'? A student might think they don't need to solve for constants. Or is it just 'style'? No, it's a missing step in the logical flow of 'solving'. However, often in these datasets, if the verification is complete, it's considered a valid 'check'. But the prompt says 'Solve'. If I submit a blank paper with just the answer and 'it works', I get no credit. The solution presented is essentially 'Here is the answer, and here is proof it works.' It does not show the work to get the answer. This is a 'fail' because it doesn't actually solve the problem, it only verifies the result. The sentence 'the two conditions fix C1 and C2' is a claim, not a derivation. The subsequent steps do not derive C1 and C2; they verify the final expression. Therefore, the solution is incomplete/misleading as a tutorial.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the characteristic roots, sets up the general solution, and verifies that the stated answer satisfies both the differential equation and the initial conditions.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The characteristic equation derived in step 1 (m^2 - m - 2 = 0) does not match the differential equation y'' - y' - 2y = 0, which yields m^2 - m - 2 = 0. Wait, let's re-read carefully. The problem is -2y - y' + y'' = 0, which is y'' - y' - 2y = 0. The characteristic equation is indeed m^2 - m - 2 = 0, with roots 2 and -1. So step 1 is correct. Step 2 gives the general solution y = C1 e^(2x) + C2 e^(-x). The stated answer is y = (e^(3x) + 11)e^(-x)/3 = 1/3 e^(2x) + 11/3 e^(-x). Let's check the initial conditions with this specific solution. y(0) = 1/3 + 11/3 = 4. Correct. y'(x) = 2/3 e^(2x) - 11/3 e^(-x). y'(0) = 2/3 - 11/3 = -9/3 = -3. Correct. The solution satisfies the ODE and ICs. Why did I think it was wrong? Let me re-read the prompt's solution steps. Step 1 says m^2 - m - 2 = (m-2)(m+1). Roots 2, -1. Correct. Step 2 says y = C1 exp(2x) + C2 exp(-x). Correct. Steps 3-5 verify the final answer. The logic is sound. Is there a subtle error? The problem asks to 'Solve'. The solution jumps from the general form to verifying a specific answer without showing the calculation of C1 and C2. However, the prompt says 'Each line... is either an EQUATION... or a SENTENCE... Check that... each conclusion follows from the lines before it'. Step 2 states the general solution. Steps 3-5 verify the specific answer. It doesn't explicitly derive C1 and C2, but it verifies the result. Is this 'error' or 'style'? The prompt asks to review 'worked calculus solutions'. A solution that just verifies the answer without showing the work to find it is often considered incomplete, but is it 'wrong'? The instructions say 'find what is wrong'. If the steps are logically valid (verification is a valid way to show a solution is correct), then it passes. But usually, 'Solve' implies finding the constants. Let's look closer. Step 2 says 'the two conditions fix C1 and C2'. It does not state what they are. Then it jumps to verifying the final answer. This is a gap in the derivation. However, is it an 'error'? No. Is it 'misleading'? Maybe. Is it 'style'? Probably. But wait, look at the stated answer: y = (e^(3x) + 11)e^(-x)/3. This simplifies to 1/3 e^(2x) + 11/3 e^(-x). The roots are 2 and -1. The terms are e^(2x) and e^(-x). Everything matches. The verification steps are correct. The only issue is the lack of derivation for the constants. In many contexts, this is acceptable if the question is 'verify' or if the answer is given. Here the problem is 'Solve'. A student might be confused why the constants are 1/3 and 11/3. But the solution is technically correct in its assertions. Let me double check the characteristic equation again. y'' - y' - 2y = 0. m^2 - m - 2 = 0. (m-2)(m+1)=0. m=2, m=-1. Correct. General solution y = c1 e^(2x) + c2 e^(-x). y(0) = c1 + c2 = 4. y'(0) = 2c1 - c2 = -3. Adding: 3c1 = 1 => c1 = 1/3. c2 = 4 - 1/3 = 11/3. The answer is correct. The solution steps are: 1. Find roots. 2. State general form. 3. Verify answer satisfies ODE. 4. Verify answer satisfies IC1. 5. Verify answer satisfies IC2. This is a valid proof that the stated answer is the solution. It is not a derivation, but it is a valid solution method (verification). Is there any error? No. Verdict: 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/second_order_ivp_bvp, checked 2026-10-18 with SymPy 1.14.0.