An analytical procedure for solving Fredholm integral equations of the second kind with Pincherle-Goursat (degenerate) kernels. In the invariant imbedding approach used, the solution at a fixed value of t is studied as the length of the interval is varied. A Cauchy problem is derived, and it is verified that the initial-value method produces a solution of the integral equation. Such a procedure should prove a valuable alternative to the usual algebraic method, and should find application in signal detection, gas dynamics, radiative transfer, and mathematical biology. 20 pp. Refs.