A Treatise on Differential Equations: Supplementary VolumeMacmillan and Company, 1865 - 496 sider |
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Resultat 1-5 av 27
Side 3
... equal to K. Hence the solution becomes du = C. In applying this theory to the reduction of the general solution ( 2 ) in the case in which m1 = m2 , it must be observed that the numerator of the first member is the same function of m1 ...
... equal to K. Hence the solution becomes du = C. In applying this theory to the reduction of the general solution ( 2 ) in the case in which m1 = m2 , it must be observed that the numerator of the first member is the same function of m1 ...
Side 6
... equal portions , to each of which the general theorem of solution may be applied . If x - x be very small the theorem may be approximately represented by y — Y。= ƒ ( x . , Y. ) ( x − x ) . On these principles Cauchy has founded ...
... equal portions , to each of which the general theorem of solution may be applied . If x - x be very small the theorem may be approximately represented by y — Y。= ƒ ( x . , Y. ) ( x − x ) . On these principles Cauchy has founded ...
Side 14
... equal quantities . [ See Chap . VIII . Art . 8. ] It would not there- fore be infinite . Hence we conclude that would not become dp dy infinite for a particular primitive in the strict sense of that term , i . e . for a solution derived ...
... equal quantities . [ See Chap . VIII . Art . 8. ] It would not there- fore be infinite . Hence we conclude that would not become dp dy infinite for a particular primitive in the strict sense of that term , i . e . for a solution derived ...
Side 30
... equal to f ( x , u ) , or to f ( x , u ) deprived of any factor which neither vanishes nor becomes infinite when u 0. If that integral tend to 0 with u the solution is singular . = Ex . 1. Determine whether y = 0 is a singular solution ...
... equal to f ( x , u ) , or to f ( x , u ) deprived of any factor which neither vanishes nor becomes infinite when u 0. If that integral tend to 0 with u the solution is singular . = Ex . 1. Determine whether y = 0 is a singular solution ...
Side 31
... equal to 0. But the course of the demonstra- tion shews that the value of the definite integral must be first obtained on the hypothesis that u ( in this case replaced by y ) is finite , and then the limiting value which its expression ...
... equal to 0. But the course of the demonstra- tion shews that the value of the definite integral must be first obtained on the hypothesis that u ( in this case replaced by y ) is finite , and then the limiting value which its expression ...
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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...