A Treatise on Differential EquationsMacmillan, 1865 - 496 páginas |
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Términos y frases comunes
2ndly arbitrary constants arbitrary function assume C₁ C₂ Chap complete primitive condition corresponding 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 dx₁ dx² dy dz dz dx dz dz eliminating equa equation dy exact differential expressed finite form dx given curve given equation Hence homogeneous functions independent variable integrating factor involving 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 shewn singular solution substituting suppose theorem tion transformation whence X₁ y₁ аф
Pasajes populares
Página 23 - N being functions of the variables x and y. In this mode of representation x is regarded as the independent variable and y as the dependent variable. We may, however, regard y as the independent and x as the dependent variable, on which supposition the form of the typical equation will be...
Página 270 - ... Shew that the above is a particular case of curves of pursuit. 6. Find the orthogonal trajectory of a system of circles touching a given straight line in a given point. 7. Find the orthogonal trajectory of the system of ellipses a;2 «* defined by the equation —2 + ~ = I, 5 being the variable Of O parameter.
Página 34 - N are homogeneous functions of x and y of the ntti degree. We may then, in accordance with the known type of homogeneous functions, write so that the equation Mdx + Ndy — 0 becomes on substitution...
Página 29 - ... a repetition of the first supposed complete primitive. Certain cases in which the equation Mdx + Ndy = 0 admits of finite solution. 4. The equation Mdx + Ndy = 0 can always be solved when the variables in M and N admit of being separated ; ie when the equation can be reduced to the form...
Página vi - ... reason for grounding the order of exposition upon the historical sequence of discovery, that by so doing we are most likely to present each new form of truth to the mind, precisely at that stage at which the mind is most fitted to receive it, or even, like that of the discoverer, to go forth to meet it. Of the many forms of false culture, a premature converse with abstractions is perhaps the most likely to prove fatal to the growth of a masculine vigour of intellect.
Página vii - ... not doubting, in so doing, the truth which lies at the bottom of his symbolism, but distrusting his own powers. The question of the true value and proper place of symbolical methods is undoubtedly of great importance. Their convenient simplicity — their condensed power — must ever constitute their first claim upon attention. I believe however that, in order to form a just estimate, we must consider them in another aspect, viz. as in some sort the visible manifestation of truths relating to...