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First-order linear equations

Problem 6.24 · medium Video

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

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 = 3 fixes the constant of integration: C = 3.
\[ \left(3 x + 9\right) e^{- 3 x} + \frac{d}{d x} \left(x + 3\right) e^{- 3 x} = e^{- 3 x} \]
The solution satisfies the equation.✓ Proved
\[ 3 \]
And the initial condition.

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

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, 3) is highlighted, y = (x + 3)·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 in line 5 is 3*(x + 3)*exp(-3*x), which does not satisfy the initial condition y(0) = 3 (it gives y(0) = 9). The stated answer (x + 3)*exp(-3*x) is correct, but the derivation contains a factor of 3 error in the final expression.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution in line 5 is 3*(x + 3)*exp(-3*x), which does not satisfy the initial condition y(0) = 3 (it gives y(0) = 9). The stated answer (x + 3)*exp(-3*x) is correct, but the derivation contains a factor of 3 error in the final expression.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution presented in step 5 is 3*(x + 3)*exp(-3*x), which does not satisfy the initial condition y(0)=3 (it yields 9). The stated answer (x + 3)*exp(-3*x) is correct, but the derivation implies C=3 leads to the wrong function, or the function shown is incorrect. Specifically, the general solution is y = (x+C)e^{-3x}, so y(0)=C=3 gives y=(x+3)e^{-3x}. Step 5 checks 3*(x+3)e^{-3x}, which is 3 times the correct answer.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution in line 5 is 3 times the correct answer. The correct general solution is y = (x + C)e^{-3x}. Applying y(0)=3 gives C=3, so y = (x+3)e^{-3x}. Line 5 checks 3(x+3)e^{-3x}, which satisfies the ODE but fails the initial condition (y(0)=9). The text in line 4 claims C=3, but the subsequent verification uses a function with an extra factor of 3, contradicting the stated answer and the initial condition.
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.