Numerical ODE: Euler, Runge–Kutta

• A.Wrong

  False.

  1.50 overshoots badly.

  The increments here are small — each step adds roughly 0.05 × 2.5 ≈ 0.125 — so two steps can move y by only about a quarter of a unit.

• B.WrongExecution slip

  False.

  1.65 is roughly what a single step of size 0.1 would give if the slope were mistakenly evaluated as $\sqrt{6} \approx 2.45$ and then multiplied by a larger step, or what four steps would give.

  The problem specifies h = 0.05 from x = 1 to x = 1.1

  which is exactly two steps.

• C.Correct

  True.

  Step $1: f(1, 1) = \sqrt{3 + 2 + 1} = \sqrt{6}$

  so $y_{1} = 1 + 0.05\sqrt{6} \approx 1.1225 ($verified).

  Step 2 at $x = 1.05: f = \sqrt{3.15 + 2\cdot1.1225 + 1} = \sqrt{6.395} \approx 2.5288$

  so $y_{2} \approx 1.1225 + 0.1264 \approx 1.2489 ($verified exactly as $1 + \sqrt{6}/20 + \sqrt{10\sqrt{6} + 615}/200)$.

  Rounded, 1.25.

  The trigger: count the steps first — (1.1 − 1)/0.05 = 2 — then update x and y together at each one.

• D.WrongExecution slip

  False.

  1.15 is close to the result of taking only the first step and stopping (1.1225 rounds to 1.12), or of using a step size half the one specified.

  The second step contributes another 0.126

  which is what separates this from the right answer.

2 of these 3 wrong options correspond to a named reasoning error — the same errors recur across subjects.