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On the solution of linear differential equations

1851; Royal Society; Volume: 5; Linguagem: Inglês

10.1098/rspl.1843.0130

2053-9134

Charles James Hargreave,

History and Theory of Mathematics

1. By the aid of two simple theorems expressing the laws under which the operations of differentiation combine with operations denoted by factors, functions of the independent variable, the author arrives at a principle extensively applicable to the solution of equations, which may be stated as follows:—“if any linear equation φ ( x , D). u = X have for its solution u = ψ ( x , D). X, this solution being so written that the operations included under the function ψ are not performed or suppressed, then φ (D, - x ). u = X has for its solution u = ψ (D, - x ). X.” The solution thus obtained may not be, and often is not, interpretable, at least in finite terms; but if by any transformation a meaning can be attached to this form, it will be found to represent a true result. An important solution immediately deducible from this principle is given by Mr. Boole in the Philosophical Magazine for February 1847, and is extensively employed in the present paper. It is immediately obtained by making the conversion above proposed in the general equation of the first order and its solution.