Home›Calculus 2›First-order linear equations›Problem 6.106

First-order linear equations

Problem 6.106 · medium Video

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

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

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

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

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 integrating the right-hand side (x) and dividing by the integrating factor to obtain the general solution y = (x + C)e^{-3x}. It jumps from setting up the integral to determining the constant C, leaving the student without the derivation of the particular solution form.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution skips the crucial step of integrating the right-hand side (x) and dividing by the integrating factor to obtain the general solution y = (x + C)e^{-3x}. It jumps from setting up the integral to determining the constant C, leaving the student without the derivation of the particular solution form.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the integrating factor, performs the integration, determines the constant from the initial condition, and verifies the result. The steps are logically sound and mathematically correct.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-26
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.