Fredholm and Volterra equations

• A.WrongExecution slip

  False.

  The denominator should be $e^{2} - 1$, not $e^{2} + 1$.

  That sign comes from evaluating $\int_{0}^{1} t^{2}e^{2t}$ dt: SymPy gives $e^{2}/4 - 1/4, a$ difference.

  Adding the boundary terms instead of subtracting produces this option.

• B.Wrong

  False on both counts — the numerator should be 4, not 2, and the denominator $e^{2} - 1$, not $e^{2} + 1$.

  The 4 comes from the ¼ in the evaluated integral.

• C.Correct

  True.

  The kernel $x^{2}e^{x+t}$ $= (x^{2}e$ˣ)(e^t) is separable

  so any solution has the form $y = c\cdot{}x^{2}e$ˣ.

  Substituting gives $c = \lambda{}c\int_{0}^{1} t^{2}e^{2t}$ dt, and for c ≠ 0 this requires $\lambda\int_{0}^{1} t^{2}e^{2t}$ dt = 1.

  SymPy evaluates that integral as $(e^{2} - 1)/4 \approx 1.5973$

  so $\lambda = 4/(e^{2} - 1) \approx 0.6261 ($verified).

  The trigger: a kernel that factors as g(x)h(t) reduces the whole problem to one scalar equation — write y = c·g(x) immediately.

• D.WrongExecution slip

  False.

  The denominator is right but the numerator is halved.

  $\int_{0}^{1} t^{2}e^{2t}$ dt $= (e^{2} - 1)/4$, and inverting a quarter gives a factor 4, not 2.

  Mis-integrating by parts once rather than twice is the usual source of the missing factor.

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