Mathematics

This problem hinges on the Leibniz integral rule for differentiating under the integral sign. The key insight is that when you differentiate $y$ twice with respect to $x$, the upper limit $x$ appears both explicitly in the integrand and implicitly through the $(x-t)$ term—you'll need to carefully apply the product rule and chain rule to both. Once you work through the derivatives, you'll discover that the second derivative naturally produces a term involving $K^2y$ plus a remainder that equals $Kf(x)$, making the differential equation fall into place.

This problem hinges on the Leibniz integral rule for differentiating under the integral sign. The key insight is that when you differentiate $y$ twice with respect to $x$, the upper limit $x$ appears both explicitly in the integrand and implicitly through the $(x-t)$ term—you'll need to carefully apply the product rule and chain rule to both. Once you work through the derivatives, you'll discover that the second derivative naturally produces a term involving $K^2y$ plus a remainder that equals $Kf(x)$, making the differential equation fall into place.