The problem of neutron slowing down in an infinite medium with energy-dependent anisotropy of elastic scattering has been discussed. The scattering function, P(u′, Δu), is redefined and expanded in terms of Legendre polynomials and the energy-dependent coefficients of the expansion are determined; in this expansion of P(u′, Δu) it is possible to carry out matrix degeneration of the kernel of the slowing-down equation; the matrix separable kernel allows the transformation of the integral equation into a differential equation in terms of Green's slowing-down functions. In some cases it is possible to obtain analytically the Green's slowing-down functions. In general, these functions are determined by standard numerical methods for solving sets of differential equations.