A Treatise on Differential Equations: With a Collection of Examples Arranged in Classes, Corresponding to the Several Divisions of the Subject

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W. P. Grant, 1832 - Differential equations - 170 pages
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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 17 - ... the sum of the indices of the variables in each term is the same for every term. For example, a-1 + + y + y4 is a rational, integral, homogeneous function of the fourth degree in x and y.
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).
Page 81 - P = f-m-*dx = therefore the equations (2) become - - ,, m — я therefore, multiplying the first by е*ж and the second by and subtracting the second from the first, we get for the complete integral.

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