A Treatise on Differential Equations, Volume 1There is an aspect of Boole's work that is not closely related to his treatises in logic or the theory of sets but which is familiar to every student of differential equations. This is the algorithm of differential operators, which he introduced to facilitate the treatment of linear differential equations. If, for example, we wish to solve the differential equation ay + by + cy = 0, the equation is written in the notation (aD2 + bD + c)y = 0. Then, regarding D as an unknown quantity rather than an operator, we solve the algebraic quadratic equation aD2 + bD + c = 0. There are many other situations in which Boole, in his Treatise on Differential Equations of 1859, pointed out parallels between the properties of the differential operator (and its inverse) and the rules of algebra. British mathematicians in the second half of the nineteenth century were thus again becoming leaders in algorithmic analysis, a field in which, fifty years earlier, they had been badly deficient. |
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2ndly arbitrary constant arbitrary function C₁ C₂ Cambridge Chap cloth complete primitive condition corresponding Crown 8vo d²u d³y deduce degree derived determined differential coefficients dp dq dt dt dv dv dx dx dx dy dy dx dz dx² dy dx dy dz dz dx dz dz eliminating equa equal exact differential expressed Fcap given equation Hence homogeneous functions independent variable integrating factor intrinsic equation involving linear equation Mdx+Ndy method obtained ordinary differential equations P₁ partial differential equation particular integral primitive equation problem proposed equation reduced relation represent respect result satisfy second member second order Shew singular solution substituting supposed symbolical theorem tion transformation V₁ whence X₁ y₁ аф