On the solution of linear differential equations

The methods employed in this paper to effect the solution or reduction of linear differential equations consist of certain peculiar transformations, and each particular class of equations is transformed by a distinct process peculiarly its own. The reduction is effected by means of certain general theorems in the calculus of operations. The terms which form the first member of the first class of equations are functions of the symbols ɯ and τ, the latter being a function ofx , and the former a function ofx and D,x being the independent variable. This member of the equations contains two arbitrary functions of vs, and may therefore be of any order whatever. It likewise contains two simple factors, such for example as ɯ+nk and which factors are taken away by the transformation employed, and consequently the equation is reduced an order lower; it is therefore integrated when of the second order. There is a series of equations of this class, each essentially distinct from the rest, yet all reducible by a similar process.