Solution of an integral equation occurring in the theories of prediction and detection

In many of the theories of prediction and detection developed during the past decade, one encounters linear integral equations which can be subsumed under the general form
\int_a^b R(t, \tau) x(\tau) d\tau = f(t), a \leqq t \leqq b
. This equation includes as special cases the Wiener-Hopf equation and the modified Wiener-Hopf equation
\int_0^T R(\mid t - \tau \mid ) x(\tau) d\tau = f(t), 0 \leqq t \leqq T
. The type of kernel considered in this note occurs when the noise can be regarded as the result of operating on white noise with a succession of not necessarily time-invariant linear differential and inverse-differential operators. For this type of noise, which is essentially a generalization of the stationary noise with a rational spectral density function, it is shown that the solution of the integral equation can be expressed in terms of solution of a certain linear differential equation with variable coefficients.