A Treatise on Differential Equations, Τόμος 1Macmillan, 1872 - 494 σελίδες There 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. |
Αναζήτηση στο βιβλίο
Αποτελέσματα 1 - 5 από τα 45.
Σελίδα xiii
... Curves of Pursuit , 251. Solution of a differential equation , 254. Involutes , 256 . Inverse problem of caustics , 258-263 . Direct problem , 259 . Intrinsic equation of a curve , 263. Exercises , 269 . 209 238 CHAPTER XII . ORDINARY ...
... Curves of Pursuit , 251. Solution of a differential equation , 254. Involutes , 256 . Inverse problem of caustics , 258-263 . Direct problem , 259 . Intrinsic equation of a curve , 263. Exercises , 269 . 209 238 CHAPTER XII . ORDINARY ...
Σελίδα 1
... curve , it defines the rate of variation of the one , y , as referred to that of the other , x , by means of the fundamental con- ception of a limit ; it expresses that ratio by a differential dy coefficient d ; and of that differential ...
... curve , it defines the rate of variation of the one , y , as referred to that of the other , x , by means of the fundamental con- ception of a limit ; it expresses that ratio by a differential dy coefficient d ; and of that differential ...
Σελίδα 12
... curve , i.e. of a curve determined as to its species by the form of its equation , and as to its elements by the values of the constants in that equation , correspond only definite values of the ordinate y determining the corresponding ...
... curve , i.e. of a curve determined as to its species by the form of its equation , and as to its elements by the values of the constants in that equation , correspond only definite values of the ordinate y determining the corresponding ...
Σελίδα 20
... curve , the higher and more absolute is the property which that differential equation expresses . We reserve to a future Chapter the consideration of the genesis of partial differential equations as well as of ordinary differential ...
... curve , the higher and more absolute is the property which that differential equation expresses . We reserve to a future Chapter the consideration of the genesis of partial differential equations as well as of ordinary differential ...
Σελίδα 28
... curve the species of which as dependent upon the form of its equation would be determined by a law , but an element of which , re- presented by a constant in that equation , would be left arbi- trary , so as to permit us to draw the curve ...
... curve the species of which as dependent upon the form of its equation would be determined by a law , but an element of which , re- presented by a constant in that equation , would be left arbi- trary , so as to permit us to draw the curve ...
Άλλες εκδόσεις - Προβολή όλων
Συχνά εμφανιζόμενοι όροι και φράσεις
2ndly algebraic arbitrary constants arbitrary function assume C₁ C₂ Chap Chapter complete primitive condition Crown 8vo curve deduce derived determined differential coefficients dp dp dp dq dp dy dt dt dv du dv dv dv dx dx dx dy dy dx dz dx² dy dx dy dz dz dx dz dy dz dz Edition eliminating equa exact differential expressed fcap finite given equation Hence homogeneous functions independent variable integrating factor involving method Mx+Ny obtained ordinary differential equations P₁ partial differential equation particular integral pdx+qdy primitive equation reduced relation represent respect result satisfied second member second order Shew shewn singular solution substituting suppose theorem tion transformation whence X₁ y₁