Discussion
The kernel of the linear integral equation
\[ x(t) = \lambda \int_{0}^{1} (t^{2} - s^{2})\, x(s)\, ds \]
is :
Answer: A
The kernel \(t^{2} - s^{2}\) is separable.
A separable (degenerate) kernel is a finite sum of products \(a_{i}(t) b_{i}(s)\). Here \(t^{2} - s^{2} = t^{2} \cdot 1 + 1 \cdot (-s^{2})\), a sum of two such products, so the integral equation reduces to a 2 × 2 linear system.
It is not symmetric: swapping \(t\) and \(s\) gives \(s^{2} - t^{2}\), the negative, so the kernel is skew-symmetric. It is not a convolution kernel, which would be a function of \(t - s\) alone, and it is bounded and continuous on the square, so it is not singular.
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