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2ndly according admit application arbitrary constant arbitrary function assume becomes Chap chapter complete primitive condition connected considered constitutes contain corresponding curve deduce definite dependent derived determined differential coefficients direct effected eliminating equa equal evident exact differential example exist expression final finite given equation gives Hence homogeneous homogeneous functions illustrate independent variable infinite integrating factor involving latter lead limit linear means method multiplying necessary observed obtained operation ordinary origin partial differential equation particular positive possible present problem proposed question rational reduced referred regarding relation represent respect result satisfied second member second order shew singular solution solve species substituting sufficient supposed symbolical theorem theory tion transformation variables whence
الصفحة 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...
الصفحة 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.
الصفحة 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...
الصفحة 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...
الصفحة 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.