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First-order linear equations

Problem 6.198 · hard

Solve \( \displaystyle y' + 2y = 2 e^{x} \) 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
\[ \int 2 e^{3 x}\, dx = \frac{2 e^{3 x}}{3} \]
Integrate the right side.✓ Proved
Setting x = 0 and y = 2 fixes the constant of integration: C = 4/3.
\[ \frac{\left(4 e^{3 x} + 8\right) e^{- 2 x}}{3} + \frac{d}{d x} \frac{\left(2 e^{3 x} + 4\right) e^{- 2 x}}{3} = 2 e^{x} \]
The solution satisfies the equation.✓ Proved
\[ 2 \]
And the initial condition.✓ Proved

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

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 skips the crucial step of integrating the product rule result and adding the constant of integration C. It jumps from the derivative form to the integral of the RHS, then abruptly states the value of C without showing the general solution y(x) = ... + C e^{-2x} or the equation used to solve for C. A student would not learn how to actually solve for the constant.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution skips the crucial step of integrating the product rule result and adding the constant of integration C. It jumps from the derivative form to the integral of the RHS, then abruptly states the value of C without showing the general solution y(x) = ... + C e^{-2x} or the equation used to solve for C. A student would not learn how to actually solve for the constant.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly identifies the integrating factor, performs the integration, and determines the constant of integration. The final verification steps confirm the solution satisfies both the differential equation and the initial condition.
gpt-oss:20b: fail (error) 2026-10-03 — The proposed solution y = 4*(e^{3x}+2)e^{-2x}/3 is incorrect; it should be y = 2*(e^{3x}+2)e^{-2x}/3. The constant C was computed correctly, but the final expression was mis‑scaled.

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 2026-10-03 with SymPy 1.14.0.