A Treatise on Differential EquationsMacmillan and Company, 1859 - 494 Seiten |
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2ndly arbitrary constant arbitrary function C₁ C₂ Chap complete primitive condition corresponding curve d²u deduce degree derived determined dF do dF differential coefficients dp dy dt dt dv dv dx dx dx dy dx dz dx² dy dy dx dy dz dz dx dz dz eliminating envelope species equa exact differential expression finite form dx given equation gives Hence homogeneous functions independent variable infinite integrating factor linear equation Mdx+Ndy method Mx+Ny obtained ordinary differential equations P₁ partial differential equation particular integral primitive equation reduced relation represent respect result satisfied second member second order shew singular solution substituting supposed symbolical form Taylor's theorem theorem tion transformation V₁ V₂ whence X₁ Y₁ аф
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Seite 82 - We thus see that in the case of homogeneous equations the employment of integrating factors Conducts us, but by a more lengthened route, to the same final integrals as the direct method of Chap. II. It is difficult to lay down any general rule as to the value of concurrent methods, but it would probably be not very remote from truth to say, that the peculiar advantage of the theory of integrating factors consists rather in its appropriateness for the investigation of conditions under which solution...
Seite 139 - DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. 1. IN the largest sense which has been given to the term, a singular solution of a differential equation is a relation between the variables which reduces the two members of the equation to an identity, but which is not included in the complete primitive. In this sense, the relation obtained by equating to 0 some common algebraic factor of the terms of the equation might claim to be called a singular solution.
Seite 23 - For as any primitive equation between x and y enables us theoretically to determine either y as a function of x, or x as a function of y, it is indifferent which of the two variables we suppose independent. It is usual...
Seite 377 - Tr"1 is defined. Thus it is the office of the inverse symbol to propose a question, not to describe an operation. It is, in its primary meaning, interrogative, not directive. Suppose the given equation to be Then on the above principle of notation we should have...
Seite 20 - Equation of lines of the second order," and attributes it to Monge, adding the words, " But here our powers of geometrical interpretation fail, and results such as this can scarcely be otherwise useful than as a registry of integrable forms.
Seite 60 - ... such a function is <£(-); therefore, finally, if M and N are homogeneous functions of x and y of a common degree. For let M and N be homogeneous and of the wth degree.