Integral representations of particular integrals of a class of inhomogeneous linear ordinary differential equations

• Published in: Journal of mathematical physics 1978-03, Vol.19 (3), p.635-641
• Main Author: Johnston, George L.
• Format: Article
• Language: English
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.523694

A method of obtaining integral representations of particular integrals of a class of inhomogeneous second‐order linear ordinary differential equations is presented. The integrands of the representations are exp[F (z;t)], where F (z;t) =z 2 a (t)+z b (t)+c (t), z is the independent variable of the differential equation, and a, b, and c are initially unspecified functions of the variable of integration, t. The lower limit of the contours of integration is zero. The upper limits of integration and the contours along which the integral is taken are initially unspecified. In the general class of inhomogeneous differential equations considered, the coefficients of the dependent variable and its derivatives are polynomials in z with complex constants and the homogeneous term is exp(k 2 z 2+k 1 z+k 0), where the k n are complex constants. One imposes the conditions that the application of the homogeneous operator to the assumed form of integral representation give the integral of −∂F/∂t, that exp[F (z;0)] be equal to the inhomogeneous term of the differential equation, and that the limit of expF as t approaches the upper limit along the contour of integration be zero. By equating coefficients of different powers of z separately to zero, one obtains a set of coupled equations for a, b, and c. The basic class of inhomogeneous differential equations to which the method is applicable is determined by requiring that a, b, and c be algebraic or elementary transcendental functions of t. The class of equations to which the method is applicable is extended to include inhomogeneous terms of the form z n exp(k 2 z 2+k 1 z+k 0), where n is a positive integer, by repeated differentiation of integral representations of members of the basic class with respect to k 1, treated as a parameter. More general inhomogeneous terms may be treated by superposition and, in appropriate cases, by approximation in terms of sets of orthogonal functions.