A Treatise on Differential Equations: Supplementary VolumeMacmillan and Company, 1865 - 496 sider |
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Side 7
... obtain an infinite system of such curves , each consisting of two branches . The aggregate of branches thus obtained is evidently the same as the aggre- gate of curves given by the two primitives ( 5 ) and ( 6 ) , un- restricted by any ...
... obtain an infinite system of such curves , each consisting of two branches . The aggregate of branches thus obtained is evidently the same as the aggre- gate of curves given by the two primitives ( 5 ) and ( 6 ) , un- restricted by any ...
Side 8
... obtain every form of solution which is implied either in the func- tional equation ( F ) , or in its component primitives V1 = c1 , V2 = C2 , Vn = cn . - The same considerations justify us in speaking of ( 15 ) as the complete primitive ...
... obtain every form of solution which is implied either in the func- tional equation ( F ) , or in its component primitives V1 = c1 , V2 = C2 , Vn = cn . - The same considerations justify us in speaking of ( 15 ) as the complete primitive ...
Side 10
... obtained would involve x only , while it is a condition accompanying the use of = ∞ dp dy that it leads to solutions involving y at least . " It is ob- jected , Why may we pass over this case ? Such a case might occur and furnish a ...
... obtained would involve x only , while it is a condition accompanying the use of = ∞ dp dy that it leads to solutions involving y at least . " It is ob- jected , Why may we pass over this case ? Such a case might occur and furnish a ...
Side 18
... obtained above . The first , though it does not determine c as a function of x , still ex- presses a relation between c and x , which is the ground of the fulfilment of the condition dp = 8 . dy We may further notice a peculiarity ...
... obtained above . The first , though it does not determine c as a function of x , still ex- presses a relation between c and x , which is the ground of the fulfilment of the condition dp = 8 . dy We may further notice a peculiarity ...
Side 20
... , but do not satisfy all the higher differential equations obtained from it , such solution will be singular and of the envelope species . Ex . 1. By comparison with its complete primitive we 20 [ CH . XXI . ADDITIONS TO CHAPTER VIII .
... , but do not satisfy all the higher differential equations obtained from it , such solution will be singular and of the envelope species . Ex . 1. By comparison with its complete primitive we 20 [ CH . XXI . ADDITIONS TO CHAPTER VIII .
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Vanlige uttrykk og setninger
arbitrary constants Chap Chapter common integral complete primitive condition Crelle's Journal deduce derived determine dF dF dF dF dp dFdF differential coefficients dp dF dp dp dx dp dq dp dy dp₁ dq dp du du du du₁ dv dv dx dp dp dx dx dx dy dx dy dy dx dz dx₁ dx² dy dp dy dy dx dy dz dz dy dz dz eliminate equa expression factor given equation Hence homogeneous function independent variables infinite Jacobi Last Multiplier linear partial differential m₁ memoir obtain ordinary differential equations P₁ partial differential equations particular integral Professor Boole proposition reduced represent respect result shewn singular solution system of ordinary theorem theory tion transformation u₁ values vanish whence x₁ y₁ аф
Populære avsnitt
Side ix - Researches on the Theory of Analytical Transformations, with a special application to the Reduction of the General Equation of the Second Order.
Side 146 - ... that the solution of the two relevant systems ultimately depends on the solution of a system of ordinary differential equations of the first order, and that from these ordinary differential equations the given equation of the second order may be deduced independently of the assumption above mentioned. 1 shall also discuss the theory of the second integration. And I shall exemplify another method of solution connected by a remarkable law of reciprocity with the above method. First Investigation,...
Side 75 - On Simultaneous Differential Equations of the First Order in which the Number of the Variables exceeds by more than one the Number of the Equations,
Side 228 - T=c,, respectively, then we have Now v being determinable by an equation of the same form as u, it follows that of the above two values of u one must be assigned to v, so that the solution of the problem will be contained in the system or in the system The particular forms of the arbitrary functions <f, and ty will depend solely upon the nature of the problem under consideration.
Side 118 - Jacobi's method by finding an integral of the first partial differential equation, a process of derivation agreeing in principle with Jacobi's, only more extended, may lead us without further integration to a point at which the discovery of a common integral of the entire system will depend only upon the solution of a single differential equation of the first order susceptible of being made integrable by a factor. Failing this, it will enable us to convert the given system of partial differential...