A Treatise on Differential Equations, Volume 1Macmillan, 1865 - 496 pages |
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Common terms and phrases
2ndly arbitrary constant arbitrary function C₁ C₂ Chap complete primitive condition corresponding d³y deduce degree derived determined differential coefficients dp dy dv du dv dv dv dx dc dx dx dx dy dy dx dz dx² dy dx dy dz dz dx dz dz eliminating envelope species equa equal exact differential expressed finite given equation gives Hence homogeneous functions independent variable infinite integrating factor involving linear equation Mdx+Ndy method multiplying 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 symbolical form theorem tion transformation V₂ whence X₁ Y₁ аф
Popular passages
Page 23 - M~ dy 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 to treat...
Page 83 - Thus a direct reference to the above theorem shews that the condition which must be satisfied in order that the equation Mdx + Ndy = 0 may...
Page 385 - TT~I 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...
Page 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.
Page 34 - To integrate a homogeneous equation it suffices to assume y = vx. In the transformed equation the variables x and v will then admit of separation. Thus in the above example we should find {vx + x...
Page 163 - ... of particular solutions, possesses a differential element common with each of them. We shall now see that this property is not accidental — that it is intimately connected with the definition of a singular solution. It is important that the two marks, positive and negative, by the union of which a singular solution of a differential equation of the first order is characterized, and by the expression of which its definition is formed, should be clearly appred*u hended.