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First-order linear equations

Problem 6.148 · medium

Solve \( \displaystyle y' + 2y = e^{- 2 x} \) with \( \displaystyle y(0) = -2 \).

The equation is linear in standard form; the integrating factor is e^(∫2 dx) = e^(2x).
\[ \frac{d}{d x} Y{\left(x \right)} e^{2 x} = 2 Y{\left(x \right)} e^{2 x} + e^{2 x} \frac{d}{d x} Y{\left(x \right)} \]
Multiplying by e^(ax) turns the left side into (e^(ax) y)'.✓ Proved
\[ \int 1\, dx = x \]
Integrate the right side.✓ Proved
Setting x = 0 and y = -2 fixes the constant of integration: C = -2.
\[ \left(2 x - 4\right) e^{- 2 x} + \frac{d}{d x} \left(x - 2\right) e^{- 2 x} = e^{- 2 x} \]
The solution satisfies the equation.✓ Proved
\[ -2 \]
And the initial condition.✓ Proved

Answer \( y = \left(x - 2\right) e^{- 2 x} \)

Lines: 4 proved, 2 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	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	Not checked	—	a sentence; read, not computed
5	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
6	✓ 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 back; y(0) matches

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the general solution derived from the integration step (Y(x) = (x + C)e^{-2x}). It jumps from integrating the right-hand side to fixing the constant, omitting the crucial algebraic step of dividing by the integrating factor and solving for y(x).

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to explicitly state the general solution derived from the integration step (Y(x) = (x + C)e^{-2x}). It jumps from integrating the right-hand side to fixing the constant, omitting the crucial algebraic step of dividing by the integrating factor and solving for y(x).
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the integrating factor, solves the differential equation, and applies the initial condition to find the constant. The final answer is correct.
gpt-oss:20b: pass 2026-09-28

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/linear_first_order, checked 2026-09-28 with SymPy 1.14.0.