Home›Calculus 3›Second-order initial and boundary value problems›Problem 6.648

Second-order initial and boundary value problems

Problem 6.648 · medium

Solve \( \displaystyle 20 y - 4 y' + y'' = 0 \) with \( \displaystyle y(0) = -2,\ y(1) = -3 \).

\[ m^{2} - 4 m + 20 = \left(m - 2 - 4 i\right) \left(m - 2 + 4 i\right) \]
The characteristic equation has complex roots 2 ± 4i.✓ Proved
So y = C₁exp(2*x)*cos(4*x) + C₂exp(2*x)*sin(4*x); the two conditions fix C₁ and C₂.
Reviewed
\[ \frac{20 \left(- 3 \sin{\left(4 x \right)} + 2 e^{2} \sin{\left(4 x - 4 \right)}\right) e^{2 x - 2}}{\sin{\left(4 \right)}} - 4 \frac{d}{d x} \frac{\left(- 3 \sin{\left(4 x \right)} + 2 e^{2} \sin{\left(4 x - 4 \right)}\right) e^{2 x - 2}}{\sin{\left(4 \right)}} + \frac{d^{2}}{d x^{2}} \frac{\left(- 3 \sin{\left(4 x \right)} + 2 e^{2} \sin{\left(4 x - 4 \right)}\right) e^{2 x - 2}}{\sin{\left(4 \right)}} = 0 \]
It satisfies the equation ...✓ Proved
\[ \left. \frac{\left(- 3 \sin{\left(4 x \right)} + 2 e^{2} \sin{\left(4 x - 4 \right)}\right) e^{2 x - 2}}{\sin{\left(4 \right)}} \right|_{\substack{ x=0 }} = -2 \]
... y(0) = -2 ...✓ Proved
\[ \left. \frac{\left(- 3 \sin{\left(4 x \right)} + 2 e^{2} \sin{\left(4 x - 4 \right)}\right) e^{2 x - 2}}{\sin{\left(4 \right)}} \right|_{\substack{ x=1 }} = -3 \]
y(1) = -3.✓ Proved

Answer \( y = \frac{\left(- 3 \sin{\left(4 x \right)} + 2 e^{2} \sin{\left(4 x - 4 \right)}\right) e^{2 x - 2}}{\sin{\left(4 \right)}} \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-16
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15
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 2026-10-19 with SymPy 1.14.0.