Home›Calculus 2›First-order linear equations›Problem 6.16

First-order linear equations

Problem 6.16 · hard Video

Solve \( \displaystyle y' + 3y = 2 e^{- 3 x} \) 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
\[ \int 2\, dx = 2 x \]
Integrate the right side.✓ Proved
Setting x = 0 and y = 1 fixes the constant of integration: C = 1.
\[ \left(6 x + 3\right) e^{- 3 x} + \frac{d}{d x} \left(2 x + 1\right) e^{- 3 x} = 2 e^{- 3 x} \]
The solution satisfies the equation.✓ Proved
\[ 1 \]
And the initial condition.

Answer \( y = \left(2 x + 1\right) e^{- 3 x} \)

The slope field of y' = -3y + 2e^(-3x): short segments showing the slope at each point. Several solution curves follow the field; the one through (0, 1) is highlighted, y = (2x + 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 after multiplying by the integrating factor. The RHS becomes 2, so the integrated form is Y*e^(3x) = 2x + C, but the solution jumps from the derivative setup to the final answer without showing this integration or the subsequent division by e^(3x).

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution skips the crucial step of integrating the right-hand side after multiplying by the integrating factor. The RHS becomes 2, so the integrated form is Y*e^(3x) = 2x + C, but the solution jumps from the derivative setup to the final answer without showing this integration or the subsequent division by e^(3x).
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. It incorrectly jumps from the derivative of the product to the integral of the original RHS (2), omitting the factor e^(3x) which should yield 2x*e^(3x). Consequently, the derived general solution is missing the homogeneous part's constant application logic, although the final answer happens to be correct by coincidence of the specific initial condition and form, the reasoning presented in steps 2-4 is mathematically invalid.
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 (2*e^(3x)) to find the general solution. It jumps from the derivative form to the final answer without showing the integration or the determination of the constant C via the general solution y = (2x + C)e^(-3x). While the final answer is correct, the logical flow is broken and does not demonstrate the method.
gpt-oss:20b: fail (error) 2026-09-26 — The integrating factor method was applied correctly, but the integral of the right‑hand side was taken as 2 instead of 2 e^{3x}. Consequently the derived solution (2x+1)e^{-3x} does not satisfy the differential equation; the correct solution is y=2/3 + C e^{-3x} with C=1/3 to meet y(0)=1.

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.