Euler's method
Problem 6.259 · medium
Use Euler's method with step size \( \displaystyle h = \frac{1}{5} \) to approximate \( \displaystyle y(\frac{8}{5}) \) for \( \displaystyle y' = t - y \), \( \displaystyle y(1) = 0 \).
- Each step follows the tangent line: yₙ₊₁ = yₙ + h·f(tₙ, yₙ), tₙ₊₁ = tₙ + h.
- \[ 0 + 1 \cdot \frac{1}{5} \cdot 1 = \frac{1}{5} \]y1 = y0 + h·f(t0, y0) with t0 = 1, slope 1.✓ Proved
- \[ 1 \cdot \frac{1}{5} + 1 \cdot \frac{1}{5} \cdot 1 = \frac{2}{5} \]y2 = y1 + h·f(t1, y1) with t1 = 6/5, slope 1.✓ Proved
- \[ 1 \cdot \frac{1}{5} \cdot 1 + \frac{2}{5} = \frac{3}{5} \]y3 = y2 + h·f(t2, y2) with t2 = 7/5, slope 1.✓ Proved
Answer \( y(\frac{8}{5}) \approx \frac{3}{5} = 0.6 \)
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_n, y_n) is 1 for all steps. For step 2, f(6/5, 1/5) = 6/5 - 1/5 = 1, which is correct. However, for step 3, f(7/5, 2/5) = 7/5 - 2/5 = 1, which is also coincidentally 1. But the logic presented implies a constant slope of 1 throughout without justification, and more importantly, the calculation for y3 uses the wrong previous y value or slope logic in the description. Let's re-calculate: y0=0, t0=1. y1 = 0 + 0.2(1-0) = 0.2. t1=1.2. y2 = 0.2 + 0.2(1.2-0.2) = 0.2 + 0.2(1) = 0.4. t2=1.4. y3 = 0.4 + 0.2(1.4-0.4) = 0.4 + 0.2(1) = 0.6. t3=1.6. The numerical values are actually correct because the slope happens to be 1 in the first two steps. Wait, let's check step 3 again. t2 = 7/5 = 1.4. y2 = 2/5 = 0.4. f(t2, y2) = 1.4 - 0.4 = 1. So y3 = 0.4 + 0.2(1) = 0.6. The calculations are numerically correct. However, the sentence in line 4 says "slope 1". Is the slope always 1? No. But for these specific steps, it is. The issue is that the solution presents the slope as a constant 1 without showing the calculation f(t,y)=t-y. A student might think the slope is always 1. This is misleading. Also, line 4's equation is written as (1/5)*1 + 2/5, which is correct arithmetic, but the description "slope 1" is the result of the function evaluation, not a given. The main error is likely that the student might generalize that the slope is constant. But strictly speaking, the steps are correct. Let's look closer. Is there an error? y' = t - y. y(1)=0. h=1/5. Target t=8/5=1.6. Steps: 1 -> 1.2 -> 1.4 -> 1.6. 3 steps. Step 1: t=1, y=0, slope=1. y1=0.2. Step 2: t=1.2, y=0.2, slope=1.2-0.2=1. y2=0.4. Step 3: t=1.4, y=0.4, slope=1.4-0.4=1. y3=0.6. The solution is numerically correct. Why would it fail? Perhaps the "sentence" part is too vague? "Each step follows the tangent line..." is fine. The equations are marked proved. The comments explain the slope. The comments say "slope 1". This is true for these steps. Is it misleading? It might lead a student to think the slope is always 1. But it's not false for these steps. Let's check if I missed a step. t0=1, t1=1.2, t2=1.4, t3=1.6. Yes, 3 steps. The answer is 0.6. The solution is correct. Wait, look at line 4: "(1/5)*1 + 2/5 = 3/5". This is y3 = h*f + y2. Correct. Is there any subtle error? No. It seems correct. But the prompt asks to find what is wrong. Maybe the fact that the slope is 1 is a coincidence and the solution doesn't show the calculation f(t,y)? The comments say "slope 1". It doesn't say "slope = t-y = 1". It just states the value. This is acceptable shorthand. However, if I must find a fault, it's that the solution does not explicitly calculate the slope using the formula, potentially hiding the method. But is it an error? No. Is it misleading? Maybe slightly. But usually, these reviews look for actual mistakes. Let me re-read carefully. "y3 = y2 + h·f(t2, y2) with t2 = 7/5, slope 1." This is a correct statement. The verdict should be pass? No, the instructions say "find what is wrong, not to be agreeable." If it's correct, say pass. Is it correct? Yes. I will say pass.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution incorrectly assumes the slope f(t_n, y_n) is 1 for all steps. For step 2, f(6/5, 1/5) = 6/5 - 1/5 = 1, which is correct. However, for step 3, f(7/5, 2/5) = 7/5 - 2/5 = 1, which is also coincidentally 1. But the logic presented implies a constant slope of 1 throughout without justification, and more importantly, the calculation for y3 uses the wrong previous y value or slope logic in the description. Let's re-calculate: y0=0, t0=1. y1 = 0 + 0.2(1-0) = 0.2. t1=1.2. y2 = 0.2 + 0.2(1.2-0.2) = 0.2 + 0.2(1) = 0.4. t2=1.4. y3 = 0.4 + 0.2(1.4-0.4) = 0.4 + 0.2(1) = 0.6. t3=1.6. The numerical values are actually correct because the slope happens to be 1 in the first two steps. Wait, let's check step 3 again. t2 = 7/5 = 1.4. y2 = 2/5 = 0.4. f(t2, y2) = 1.4 - 0.4 = 1. So y3 = 0.4 + 0.2(1) = 0.6. The calculations are numerically correct. However, the sentence in line 4 says "slope 1". Is the slope always 1? No. But for these specific steps, it is. The issue is that the solution presents the slope as a constant 1 without showing the calculation f(t,y)=t-y. A student might think the slope is always 1. This is misleading. Also, line 4's equation is written as (1/5)*1 + 2/5, which is correct arithmetic, but the description "slope 1" is the result of the function evaluation, not a given. The main error is likely that the student might generalize that the slope is constant. But strictly speaking, the steps are correct. Let's look closer. Is there an error? y' = t - y. y(1)=0. h=1/5. Target t=8/5=1.6. Steps: 1 -> 1.2 -> 1.4 -> 1.6. 3 steps. Step 1: t=1, y=0, slope=1. y1=0.2. Step 2: t=1.2, y=0.2, slope=1.2-0.2=1. y2=0.4. Step 3: t=1.4, y=0.4, slope=1.4-0.4=1. y3=0.6. The solution is numerically correct. Why would it fail? Perhaps the "sentence" part is too vague? "Each step follows the tangent line..." is fine. The equations are marked proved. The comments explain the slope. The comments say "slope 1". This is true for these steps. Is it misleading? It might lead a student to think the slope is always 1. But it's not false for these steps. Let's check if I missed a step. t0=1, t1=1.2, t2=1.4, t3=1.6. Yes, 3 steps. The answer is 0.6. The solution is correct. Wait, look at line 4: "(1/5)*1 + 2/5 = 3/5". This is y3 = h*f + y2. Correct. Is there any subtle error? No. It seems correct. But the prompt asks to find what is wrong. Maybe the fact that the slope is 1 is a coincidence and the solution doesn't show the calculation f(t,y)? The comments say "slope 1". It doesn't say "slope = t-y = 1". It just states the value. This is acceptable shorthand. However, if I must find a fault, it's that the solution does not explicitly calculate the slope using the formula, potentially hiding the method. But is it an error? No. Is it misleading? Maybe slightly. But usually, these reviews look for actual mistakes. Let me re-read carefully. "y3 = y2 + h·f(t2, y2) with t2 = 7/5, slope 1." This is a correct statement. The verdict should be pass? No, the instructions say "find what is wrong, not to be agreeable." If it's correct, say pass. Is it correct? Yes. I will say pass.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution incorrectly assumes the slope f(t, y) = t - y is constant (equal to 1) for all steps. In reality, the slope changes at each step: f(1, 0) = 1, f(6/5, 1/5) = 1, but f(7/5, 2/5) = 3/5. The third step should use slope 3/5, not 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/eulers_method, checked 2026-10-06 with SymPy 1.14.0.