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First-order linear equations

Problem 6.196 · medium

Solve \( \displaystyle y' + 1y = 2 e^{x} \) with \( \displaystyle y(0) = 1 \).

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

Answer \( y = e^{x} \)

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 incorrectly integrates the right-hand side as exp(2*x) instead of 2*exp(x), leading to the wrong general solution. Although the final answer y = e^x is correct, the derivation contains a fatal algebraic error in the integration step.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution incorrectly integrates the right-hand side as exp(2*x) instead of 2*exp(x), leading to the wrong general solution. Although the final answer y = e^x is correct, the derivation contains a fatal algebraic error in the integration step.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution skips the crucial step of integrating the right-hand side and adding the constant of integration. It jumps from the derivative form directly to a specific integral result without showing the general solution y(x) = e^x + Ce^{-x}, making the determination of C in step 4 logically disconnected from the previous steps.
gpt-oss:20b: pass 2026-10-03

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.