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 39
... Hence Spax Q. dx Therefore C = ! = √ εSP2 Qdx + C , .. c being an arbitrary constant . Substituting this generalized value of C'in ( 22 ) , we have finally y = eTrix ( Se Srix Qdx + the solution required . c ) ( 23 ) , It will be ...
... Hence Spax Q. dx Therefore C = ! = √ εSP2 Qdx + C , .. c being an arbitrary constant . Substituting this generalized value of C'in ( 22 ) , we have finally y = eTrix ( Se Srix Qdx + the solution required . c ) ( 23 ) , It will be ...
Σελίδα 40
... Hence Spai Q = ( x + 1 ) 3 . x + 1 ' € Srax = ( x [ Pdx = -2 log ( x + 1 ) , ( 28 + 1 ) ~ " . Je Qdx = + 2 ( x 1 ) 2 + c . 1 ) " + c } . Setra Que = f ( x + 1 ) dx ( ( x 1 ) 2 y = ( x + 1 ) ( * + Therefore Ex . 2. Given dy ny dx x + 1 = 2 ...
... Hence Spai Q = ( x + 1 ) 3 . x + 1 ' € Srax = ( x [ Pdx = -2 log ( x + 1 ) , ( 28 + 1 ) ~ " . Je Qdx = + 2 ( x 1 ) 2 + c . 1 ) " + c } . Setra Que = f ( x + 1 ) dx ( ( x 1 ) 2 y = ( x + 1 ) ( * + Therefore Ex . 2. Given dy ny dx x + 1 = 2 ...
Σελίδα 43
... Hence M and N being given , the expressions for are implicitly given also . dy d'y Now dx ' dx2 > dy dy dx ' dx2 & c ... Hence & ( x ) = y 。· Again , o ' ( x ) is what dy dp ( x ) i . e . becomes when x = x 。• > dx dx ' Hence ' ( x ) ...
... Hence M and N being given , the expressions for are implicitly given also . dy d'y Now dx ' dx2 > dy dy dx ' dx2 & c ... Hence & ( x ) = y 。· Again , o ' ( x ) is what dy dp ( x ) i . e . becomes when x = x 。• > dx dx ' Hence ' ( x ) ...
Σελίδα 51
... Hence we find dM - -y - x dy √ ( x2 + y2 ) ) y x Y y√ ( x2 + y2 ) * dN dy ̄ ̄ ( x2 + y2 ) $ ̄ ̄ dx To obtain the complete integral we will on this occasion employ directly the general form of solution ( 12 ) . We have Hence N- ― [ Mdx ...
... Hence we find dM - -y - x dy √ ( x2 + y2 ) ) y x Y y√ ( x2 + y2 ) * dN dy ̄ ̄ ( x2 + y2 ) $ ̄ ̄ dx To obtain the complete integral we will on this occasion employ directly the general form of solution ( 12 ) . We have Hence N- ― [ Mdx ...
Σελίδα 58
... Hence the general form of the integrating factor of the equa- tion is 1 x3y3 f ( +2 ) + 3/4 ) . 4. From the typical form of the integrating factor of the equation Mdx + Ndy = 0 , it follows that if we know two par- ticular integrating ...
... Hence the general form of the integrating factor of the equa- tion is 1 x3y3 f ( +2 ) + 3/4 ) . 4. From the typical form of the integrating factor of the equation Mdx + Ndy = 0 , it follows that if we know two par- ticular integrating ...
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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