Existence–uniqueness, Picard, Lipschitz

• A.Correct

  True.

  Try $y = x^{2}$: SymPy substitutes it into $x^{2}y'' - 2x^{2}y' + (4x - 2)y$ and gets exactly 0

  so $x^{2}$ is a solution.

  It also satisfies y(0) = 0 and $\varphi(1) = 1$

  so $\varphi(x) = x^{2}$ and $\varphi(4) = 16 ($verified).

  The trigger: with polynomial coefficients of mixed degree, test low-degree monomials before attempting a series — here the degrees balance only for $x^{2}$.

• B.WrongExecution slip

  False: $\varphi(2) = 2^{2} = 4$, not 2.

  This option (and (D)) is what you get by assuming $\varphi(x) = x$

  which fits $\varphi(1) = 1$ but solves nothing — SymPy substitutes y = x and obtains 2x(x − 1)

  which is not identically zero.

  Matching the single condition $\varphi(1) = 1$ is not the same as solving the equation.

• C.Correct

  True.

  $\varphi(5) = 5^{2} = 25 ($verified).

  Note that (A) and (C) are consistent with each other and with $\varphi(x) = x^{2}$

  while (B) and (D) are consistent with the wrong guess $\varphi(x) = x$ — the four options split cleanly into two families, and identifying $\varphi$ decides both at once.

• D.WrongExecution slip

  False: $\varphi(3) = 3^{2} = 9$, not 3.

  Same underlying error as (B) — the linear guess $\varphi(x) = x$.

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