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Separable differential equations

Problem 6.120 · medium

Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - 3 y,\ y(0) = 4 \).

Separate the variables, putting every y on one side and every x on the other, and integrate both sides.
Reviewed
Use y(0) to find the constant of integration, then solve for y.
Reviewed
\[ \frac{d}{d x} 4 e^{- 3 x} = - 12 e^{- 3 x} \]
The solution satisfies the equation.✓ Proved
\[ 4 \]
And the initial condition.✓ Proved

Answer \( y = 4 e^{- 3 x} \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
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
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	sympy.checkodesol substitutes the solution into the equation; y(0) matches

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly outlines the method of separation of variables and verifies the result against both the differential equation and the initial condition.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly outlines the method of separation of variables and verifies the result against both the differential equation and the initial condition.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-27 — The solution steps are logically sound and lead to the correct answer. The verification steps confirm the result satisfies both the differential equation and the initial condition.
gpt-oss:20b: pass 2026-09-27

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/separable_ode, checked 2026-09-27 with SymPy 1.14.0.