Undetermined coefficients
Problem 6.366 · hard
Solve \( \displaystyle y'' - y = x + 2 \) with \( \displaystyle y(0) = -3 \), \( \displaystyle y'(0) = 0 \).
- \[ r^{2} - 1 = \left(r - 1\right) \left(r + 1\right) \]The characteristic equation has roots 1 and -1.✓ Proved
- So y_h = C₁e^(1x) + C₂e^(-1x). Guess y_p = A*x + B.
- \[ x + \frac{d^{2}}{d x^{2}} \left(- x - 2\right) + 2 = x + 2 \]Matching coefficients gives y_p = -x - 2; it satisfies the equation.✓ Proved
- \[ \left[\begin{matrix}\left. - x - 2 - e^{- x} \right|_{\substack{ x=0 }}\\\left. \frac{d}{d x} \left(- x - 2 - e^{- x}\right) \right|_{\substack{ x=0 }}\end{matrix}\right] = \left[\begin{matrix}-3\\0\end{matrix}\right] \]The initial conditions fix C₁ = 0 and C₂ = -1.✓ Proved
Answer \( y = - x - 2 - e^{- x} \)
Lines: 3 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.
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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | SymPy's dsolve with the same initial conditions agrees |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution incorrectly assumes C1=0 without justification, ignoring the homogeneous term C1*e^x. The stated answer y = -x - 2 - e^{-x} does not satisfy the initial condition y(0)=-3 (it gives -3, wait, -0-2-1 = -3, ok) but y'(0) = -1 - (-1) = 0? y' = -1 + e^{-x}, y'(0) = 0. So the stated answer actually satisfies the ICs. However, the general solution is y = C1 e^x + C2 e^{-x} - x - 2. With y(0)=-3 => C1+C2-2=-3 => C1+C2=-1. y'(0)=0 => C1-C2=0 => C1=C2. So 2C1=-1 => C1=-1/2, C2=-1/2. The correct solution is y = -1/2 e^x - 1/2 e^{-x} - x - 2. The stated answer y = -x - 2 - e^{-x} implies C1=0, C2=-1, which gives y(0)=-3 but y'(0) = -1 - (-1) = 0? No, y' = -1 + e^{-x}, y'(0)=0. Wait. If C1=0, C2=-1, y = -e^{-x} - x - 2. y' = e^{-x} - 1. y'(0) = 1-1=0. y(0) = -1-2=-3. So the stated answer IS a solution? Let's check the ODE: y'' - y = (e^{-x}) - (-e^{-x} - x - 2) = e^{-x} + e^{-x} + x + 2 = 2e^{-x} + x + 2. This is NOT x+2. The term 2e^{-x} remains. So the stated answer is WRONG. The error is in step 4 claiming C1=0, C2=-1 is the solution, or step 3 implying the particular solution alone is the answer. The logic in step 4 is flawed because it checks the stated answer against ICs but ignores that the stated answer is not the general solution with those constants. Actually, step 4 checks if the specific function y = -x - 2 - e^{-x} satisfies the ICs. It does. But it doesn't satisfy the ODE. The solution fails to include the e^x term required to cancel the homogeneous part generated by the particular solution's interaction or simply finds the wrong constants. The correct constants are C1=C2=-1/2. The solution presents an incorrect final answer and justifies it with a check that only verifies ICs, not the ODE.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution incorrectly assumes C1=0 without justification, ignoring the homogeneous term C1*e^x. The stated answer y = -x - 2 - e^{-x} does not satisfy the initial condition y(0)=-3 (it gives -3, wait, -0-2-1 = -3, ok) but y'(0) = -1 - (-1) = 0? y' = -1 + e^{-x}, y'(0) = 0. So the stated answer actually satisfies the ICs. However, the general solution is y = C1 e^x + C2 e^{-x} - x - 2. With y(0)=-3 => C1+C2-2=-3 => C1+C2=-1. y'(0)=0 => C1-C2=0 => C1=C2. So 2C1=-1 => C1=-1/2, C2=-1/2. The correct solution is y = -1/2 e^x - 1/2 e^{-x} - x - 2. The stated answer y = -x - 2 - e^{-x} implies C1=0, C2=-1, which gives y(0)=-3 but y'(0) = -1 - (-1) = 0? No, y' = -1 + e^{-x}, y'(0)=0. Wait. If C1=0, C2=-1, y = -e^{-x} - x - 2. y' = e^{-x} - 1. y'(0) = 1-1=0. y(0) = -1-2=-3. So the stated answer IS a solution? Let's check the ODE: y'' - y = (e^{-x}) - (-e^{-x} - x - 2) = e^{-x} + e^{-x} + x + 2 = 2e^{-x} + x + 2. This is NOT x+2. The term 2e^{-x} remains. So the stated answer is WRONG. The error is in step 4 claiming C1=0, C2=-1 is the solution, or step 3 implying the particular solution alone is the answer. The logic in step 4 is flawed because it checks the stated answer against ICs but ignores that the stated answer is not the general solution with those constants. Actually, step 4 checks if the specific function y = -x - 2 - e^{-x} satisfies the ICs. It does. But it doesn't satisfy the ODE. The solution fails to include the e^x term required to cancel the homogeneous part generated by the particular solution's interaction or simply finds the wrong constants. The correct constants are C1=C2=-1/2. The solution presents an incorrect final answer and justifies it with a check that only verifies ICs, not the ODE.qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution skips the step of determining the constants C1 and C2 for the homogeneous part. It assumes C1=0 and C2=-1 without showing the system of equations derived from y(0) and y'(0) applied to the general solution y = C1*e^x + C2*e^-x - x - 2.gpt-oss:20b: pass 2026-10-11
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/undetermined_coefficients, checked 2026-10-11 with SymPy 1.14.0.