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First-order linear equations

Problem 6.19 · hard Video

Solve \( \displaystyle y' + 4y = 3 e^{- 4 x} \) with \( \displaystyle y(0) = -1 \).

The equation is linear in standard form; the integrating factor is e^(∫4 dx) = e^(4x).
\[ \frac{d}{d x} Y{\left(x \right)} e^{4 x} = 4 Y{\left(x \right)} e^{4 x} + e^{4 x} \frac{d}{d x} Y{\left(x \right)} \]
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 = -1 fixes the constant of integration: C = -1.
\[ \left(12 x - 4\right) e^{- 4 x} + \frac{d}{d x} \left(3 x - 1\right) e^{- 4 x} = 3 e^{- 4 x} \]
The solution satisfies the equation.✓ Proved
\[ -1 \]
And the initial condition.

Answer \( y = \left(3 x - 1\right) e^{- 4 x} \)

The slope field of y' = -4y + 3e^(-4x): short segments showing the slope at each point. Several solution curves follow the field; the one through (0, -1) is highlighted, y = (3x - 1)·e^(-4x).

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^(-4x) before applying the initial condition. While the final constant C = -1 is correct, the logical jump from the integrated form to the specific solution without explicitly stating the general form is a significant gap in the derivation.

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^(-4x) before applying the initial condition. While the final constant C = -1 is correct, the logical jump from the integrated form to the specific solution without explicitly stating the general form is a significant gap in the derivation.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution skips the crucial step of writing the general solution Y(x) = (3x + C)e^{-4x} before applying the initial condition. While the final answer is correct, the logical jump from integrating the RHS to fixing the constant without stating the general form is mathematically incomplete and misleading for a student learning the method.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution skips the crucial step of integrating the right-hand side (3) to get 3x + C and combining it with the integrating factor to form the general solution y = (3x + C)e^{-4x}. It jumps directly to fixing C without showing the derivation of the particular solution structure, making the logic incomplete.
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.