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Treatise on Differential Equations: With a Collection of Examples, Arranged ...
W. C. Ottley
No preview available - 2016
arbitrary constant arbitrary functions assume complete differential complete integral complete primitive complete solution containing criterion of integrability Differential Calculus differential coefficients differentiating with respect dividing dºy dz dz eliminated equa equal equation becomes equation dy equation for determining exact differential given equation gives the integral Hence homogeneous independent variable indeterminate integrate the equation integrating factor involving an arbitrary linear equation method multiplying obtain pala partial differential equation particular integrals perfect differential proposed equation quantities satisfy the criterion satisfy the differential satisfy the equation SECT shew shewn similarly singular solution ſº solved substituting these values suppose supposition Taylor's theorem tion total differential equations transformation whence
Page 131 - ... as a summation of the terms corresponding to the elements of a single column ; this is followed by a summation of the different columns. The double integral (5), which figures in formula (2), may be evaluated by an iterated integral, either in rectangular or polar coordinates. If (5) is evaluated by integrating first with respect to y, and then with respect to x, the final formula for the triple integral is...
Page 109 - Now, multiplying the first of these equations by P, the second by — Q, and the third by R, and adding...
Page 23 - If n be not = — 2, nor = — 4, by continually repeating the same transformation as the last, the equation may successively be reduced to a series of equations of the same form as the given one, and in which the exponent of the variable becomes successively equal to m + 4 3m + 8 5m + 12 _ 7m 4. 16 ~ т~+У ~ 2m + 5...
Page 3 - Shew that every differential equation of the nth order has n first integrals. T . dx d& Integrate aVaii^P' i-^.cos'.:xy"dy + yidx = a2dx, ~ ° ~~ * F elimillate * supposing x, y, t, ~ . 1?
Page 96 - This is a linear equation of the first order, and may be integrated. But in order that the relation (4) may hold, me must have, at the same time, Q + Q'6_N + N'0 ^/N+N'0\ P + P0~M + M'0...
Page 117 - I indicates that z is to be considered as a function of the independent variables x and y and that the derivative is taken with respect to x with y held constant.
Page 104 - Equations, or those involving one or more of the partial differential coefficients of the dependent variable.
Page 96 - Multiply the second by 0 an indeterminate function of t, and adding the product to the first, we get ¡(M + M'0) z + (N + N'0) x\ dt+ (P + P'0)cfe + or (M + M'0).