V. Mechanical integration of the linear differential equations of the second order with variable coefficients

Abstract

Abstract Every linear differential equation of the second order may, as is known, be reduced to the form d/dx (1/P du/dx) = u, . . . . . . (1) where P is any given function of x. On account of the great importance of this equation in mathematical physics (vibrations of a non-uniform stretched cord, of a hanging chain, water in a canal of non-uniform breadth and depth, of air in a pipe of non-uniform sectional area, conduction of heat along a bar of non-uniform fiction or non-uniform conductivity, Laplace’s differential equation of the tides, &c. &c.), I have long endeavoured to obtain a means of faciliiting its practical solution.