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First-order linear equations

Problem 6.26 · hard Video

Solve \( \displaystyle y' + 2y = 3 e^{- 2 x} \) 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
\[ \int 3\, dx = 3 x \]
Integrate the right side.✓ Proved
Setting x = 0 and y = -2 fixes the constant of integration: C = -2.
\[ \left(6 x - 4\right) e^{- 2 x} + \frac{d}{d x} \left(3 x - 2\right) e^{- 2 x} = 3 e^{- 2 x} \]
The solution satisfies the equation.✓ Proved
\[ -2 \]
And the initial condition.

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

The slope field of y' = -2y + 3e^(-2x): short segments showing the slope at each point. Several solution curves follow the field; the one through (0, -2) is highlighted, y = (3x - 2)·e^(-2x).

Lines: 3 proved, 3 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	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	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
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 skips the crucial step of writing the general solution Y(x) = (3x + C)e^{-2x} before applying the initial condition. While the final constant C=-2 is correct, the sentence claiming that setting x=0 and y=-2 fixes C does not follow from the previous lines because the general solution was never explicitly derived or stated.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution skips the crucial step of writing the general solution Y(x) = (3x + C)e^{-2x} before applying the initial condition. While the final constant C=-2 is correct, the sentence claiming that setting x=0 and y=-2 fixes C does not follow from the previous lines because the general solution was never explicitly derived or stated.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution skips the crucial step of integrating the right-hand side after multiplying by the integrating factor. The integral of 3*e^(2x) is (3/2)e^(2x), not 3x. Consequently, the derived particular solution is incorrect, although the final answer happens to be correct by coincidence or hidden steps not shown.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution skips the crucial step of writing the general solution with the constant of integration C after integrating. It jumps from integrating the RHS to fixing C, omitting the expression Y(x)*e^(2x) = 3x + C, which is necessary to derive the final answer.
gpt-oss:20b: pass 2026-09-26

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 2026-10-13 with SymPy 1.14.0.