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. |
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Σελίδα 7
... arbitrary constant , c , by giving particular values to which a series of particular solutions is obtained . The equations xy = sin x , xy sin x + 1 , = are particular solutions of the given differential equation . The term solution is ...
... arbitrary constant , c , by giving particular values to which a series of particular solutions is obtained . The equations xy = sin x , xy sin x + 1 , = are particular solutions of the given differential equation . The term solution is ...
Σελίδα 8
... arbitrary constant c . Differentiating on the supposition that x is the independent variable , we obtain a new equation which must involved , and which may involve dx ' any or all of the quantities x , y and c . If it do not involve c ...
... arbitrary constant c . Differentiating on the supposition that x is the independent variable , we obtain a new equation which must involved , and which may involve dx ' any or all of the quantities x , y and c . If it do not involve c ...
Σελίδα 9
... arbitrary constant c be reduced to the form ( x , y ) = c , the corresponding differential equation will be obtained by mere differentiation and removal of irrelevant factors , i.e. of factors which do not contain dy , and do not ...
... arbitrary constant c be reduced to the form ( x , y ) = c , the corresponding differential equation will be obtained by mere differentiation and removal of irrelevant factors , i.e. of factors which do not contain dy , and do not ...
Σελίδα 10
... arbitrary , the value which it bears in the primitive being determined by ... constant c , we can by differentiation , and elimination ( if necessary ) of ... arbitrary constants being given , if we differentiate twice , and eliminate ...
... arbitrary , the value which it bears in the primitive being determined by ... constant c , we can by differentiation , and elimination ( if necessary ) of ... arbitrary constants being given , if we differentiate twice , and eliminate ...
Σελίδα 11
... constant b . Differentiating this equation we have x d'y dx2 = 2ax , and ... constants are eliminated does not affect the form of the final differential ... arbitrary constants consists only in this , that it is thus made to stand as ...
... constant b . Differentiating this equation we have x d'y dx2 = 2ax , and ... constants are eliminated does not affect the form of the final differential ... arbitrary constants consists only in this , that it is thus made to stand as ...
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Συχνά εμφανιζόμενοι όροι και φράσεις
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