A Treatise on Differential Equations, Volume 1 |
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2ndly arbitrary constant arbitrary function C₁ C₂ Chap complete primitive condition d³y deduce degree derived determined differential coefficients dp dq dp dy dv du dv dv dv dx dx dx dy dx dx dz dx² dy dc dy dx dy dy dy dz dz dx dz dz eliminating envelope species equa equation dy exact differential expressed finite given curve given equation Hence homogeneous functions independent variable infinite integrating factor involving linear equation Mdx+Ndy method Mx+Ny obtained ordinary differential equations partial differential equation particular integral primitive equation Prop reduced relation represent respect result satisfied second member second order shew shewn singular solution substituting suppose Taylor's theorem theorem tion transformation whence X₁ Y₁ аф
Popular passages
Page 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...
Page 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.
Page 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...
Page 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...
Page 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.
Page 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...