Euler's method
Problem 6.380 · medium
Use Euler's method with step size \( \displaystyle h = \frac{1}{4} \) to approximate \( \displaystyle y(\frac{3}{4}) \) for \( \displaystyle y' = t + y \), \( \displaystyle y(0) = -1 \).
- Each step follows the tangent line: yₙ₊₁ = yₙ + h·f(tₙ, yₙ), tₙ₊₁ = tₙ + h.
- \[ -1 + 1 \cdot \frac{1}{4} \left(-1\right) = - \frac{5}{4} \]y1 = y0 + h·f(t0, y0) with t0 = 0, slope -1.✓ Proved
- \[ - \frac{5}{4} + 1 \cdot \frac{1}{4} \left(-1\right) = - \frac{3}{2} \]y2 = y1 + h·f(t1, y1) with t1 = 1/4, slope -1.✓ Proved
- \[ - \frac{3}{2} + 1 \cdot \frac{1}{4} \left(-1\right) = - \frac{7}{4} \]y3 = y2 + h·f(t2, y2) with t2 = 1/2, slope -1.✓ Proved
Answer \( y(\frac{3}{4}) \approx - \frac{7}{4} = -1.75 \)
Lines: 3 proved, 1 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 | ✓ 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 | the steps rerun in floating point |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution incorrectly assumes the slope f(t, y) is constant (-1) for all steps. The slope depends on both t and y, so it changes at each step (e.g., at step 2, t=1/4 and y=-5/4, so slope is -1, but at step 3, t=1/2 and y=-3/2, so slope is -1, wait, let's recheck. f(t,y) = t+y. Step 1: t0=0, y0=-1, f=-1. y1 = -1 + 0.25(-1) = -1.25. t1=0.25. Step 2: t1=0.25, y1=-1.25, f = 0.25 - 1.25 = -1. y2 = -1.25 + 0.25(-1) = -1.5. t2=0.5. Step 3: t2=0.5, y2=-1.5, f = 0.5 - 1.5 = -1. y3 = -1.5 + 0.25(-1) = -1.75. Actually, the slopes happen to be -1 for the first three steps. Let's check the next one just in case. t3=0.75, y3=-1.75, f = 0.75 - 1.75 = -1. It seems the slope is always -1? Let's check the exact solution. y' - y = t. Integrating factor e^-t. (ye^-t)' = te^-t. ye^-t = -te^-t - e^-t + C. y = -t - 1 + Ce^t. y(0) = -1 => -1 = -1 + C => C=0. So y(t) = -t - 1. Then y' = -1. And t+y = t + (-t-1) = -1. So indeed y' = -1 always. The solution is correct. Why did I think it was wrong? The slopes are constant because the particular solution lies on a line where t+y is constant. Wait, t+y = -1 is constant. So f(t,y) = -1 everywhere on the solution curve. So the calculation is correct. The sentence says 'Each step follows the tangent line...'. This is a correct description of Euler's method. The calculations are correct. The answer is correct. Is there any misleading part? No. It is sound.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution incorrectly assumes the slope f(t, y) is constant (-1) for all steps. The slope depends on both t and y, so it changes at each step (e.g., at step 2, t=1/4 and y=-5/4, so slope is -1, but at step 3, t=1/2 and y=-3/2, so slope is -1, wait, let's recheck. f(t,y) = t+y. Step 1: t0=0, y0=-1, f=-1. y1 = -1 + 0.25(-1) = -1.25. t1=0.25. Step 2: t1=0.25, y1=-1.25, f = 0.25 - 1.25 = -1. y2 = -1.25 + 0.25(-1) = -1.5. t2=0.5. Step 3: t2=0.5, y2=-1.5, f = 0.5 - 1.5 = -1. y3 = -1.5 + 0.25(-1) = -1.75. Actually, the slopes happen to be -1 for the first three steps. Let's check the next one just in case. t3=0.75, y3=-1.75, f = 0.75 - 1.75 = -1. It seems the slope is always -1? Let's check the exact solution. y' - y = t. Integrating factor e^-t. (ye^-t)' = te^-t. ye^-t = -te^-t - e^-t + C. y = -t - 1 + Ce^t. y(0) = -1 => -1 = -1 + C => C=0. So y(t) = -t - 1. Then y' = -1. And t+y = t + (-t-1) = -1. So indeed y' = -1 always. The solution is correct. Why did I think it was wrong? The slopes are constant because the particular solution lies on a line where t+y is constant. Wait, t+y = -1 is constant. So f(t,y) = -1 everywhere on the solution curve. So the calculation is correct. The sentence says 'Each step follows the tangent line...'. This is a correct description of Euler's method. The calculations are correct. The answer is correct. Is there any misleading part? No. It is sound.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution incorrectly assumes the slope f(t, y) = t + y is constant (-1) for all steps. In reality, the slope changes at each step: f(0, -1) = -1, f(1/4, -5/4) = -1, but f(1/2, -3/2) = -1. Wait, let me re-calculate. Step 1: t0=0, y0=-1. f(0,-1) = -1. y1 = -1 + 0.25(-1) = -1.25. t1 = 0.25. Step 2: t1=0.25, y1=-1.25. f(0.25, -1.25) = 0.25 - 1.25 = -1. y2 = -1.25 + 0.25(-1) = -1.5. t2 = 0.5. Step 3: t2=0.5, y2=-1.5. f(0.5, -1.5) = 0.5 - 1.5 = -1. y3 = -1.5 + 0.25(-1) = -1.75. t3 = 0.75. Actually, the calculation is correct because the specific values happen to yield a slope of -1 at each step. Let me double check. y' = t+y. (0, -1) -> slope -1. (0.25, -1.25) -> 0.25 - 1.25 = -1. (0.5, -1.5) -> 0.5 - 1.5 = -1. The slopes are indeed all -1. The solution is numerically correct. However, the sentence in step 2, 3, 4 says 'slope -1' without explaining why it stays -1 or checking the new t,y values. It implies the slope is constant because y is constant or something, which is misleading reasoning even if the numbers work out by coincidence of the linear function passing through these specific points. Wait, is it a coincidence? If y(t) = t - 1, then y' = 1. But t+y = t + t - 1 = 2t - 1. Let's check the exact solution. y' - y = t. Integrating factor e^-t. (ye^-t)' = te^-t. ye^-t = -te^-t - e^-t + C. y = -t - 1 + Ce^t. y(0) = -1 => -1 = -1 + C => C=0. So y(t) = t - 1. Then y(3/4) = 3/4 - 1 = -1/4 = -0.25. Euler's method gave -1.75. Let's re-evaluate the slopes. Step 1: t0=0, y0=-1. f(0,-1) = 0 + (-1) = -1. y1 = -1 + 0.25(-1) = -1.25. Step 2: t1=0.25, y1=-1.25. f(0.25, -1.25) = 0.25 + (-1.25) = -1. y2 = -1.25 + 0.25(-1) = -1.5. Step 3: t2=0.5, y2=-1.5. f(0.5, -1.5) = 0.5 + (-1.5) = -1. y3 = -1.5 + 0.25(-1) = -1.75. The calculations are correct. The slopes are indeed all -1. Why? Because the points (0, -1), (0.25, -1.25), (0.5, -1.5) all lie on the line y = -4t - 1? No. (0, -1). (0.25, -1.25). Slope between them is (-1.25 - -1)/0.25 = -0.25/0.25 = -1. (0.5, -1.5). Slope between them is (-1.5 - -1.25)/0.25 = -0.25/0.25 = -1. The Euler steps land on points where t+y = -1. t + y = -1 => y = -t - 1. If y = -t - 1, then y' = -1. And t+y = -1. So y' = t+y holds. So the exact solution is y = t - 1? No. If y = -t - 1, y' = -1. t+y = t - t - 1 = -1. So y' = t+y is satisfied. Initial condition y(0) = -1. So the exact solution is y(t) = -t - 1. Then y(3/4) = -3/4 - 1 = -1.75. So Euler's method gives the exact answer here because the solution is linear and the step size lands on the line. The solution is correct. The notes in the equations are slightly brief but not wrong. The sentence in step 1 is correct. Wait, let me look closer at the prompt's provided solution text. Step 2: "slope -1". Step 3: "slope -1". Step 4: "slope -1". This is factually correct for this specific problem instance. Is there any error? The problem asks for y(3/4). Steps: 0 -> 1/4 (y1) 1/4 -> 2/4 (y2) 2/4 -> 3/4 (y3) So y3 is the approximation for y(3/4). The calculation is -7/4. This is correct. Verdict: pass.gpt-oss:20b: pass 2026-10-11
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/eulers_method, checked 2026-10-11 with SymPy 1.14.0.