This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1897 Excerpt: ...is seen that the general integral (5) contains only one arbitrary constant, yo. In Chapter X. we shall see that the general integral of a differential equation of the m" order may be similarly expressed by an infinite series. 73. In the following examples of differential equations of the first order to be integrated, ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1897 Excerpt: ...is seen that the general integral (5) contains only one arbitrary constant, yo. In Chapter X. we shall see that the general integral of a differential equation of the m" order may be similarly expressed by an infinite series. 73. In the following examples of differential equations of the first order to be integrated, the test for an exact differential equation should first be applied. It will be remembered that the equation Xdy-Ydx=0 is exact, if VX=JdY. dx dy, and the integral may be found by a quadrature in the form f Ydx-(X+1-f Ydx) dy = const. If the given differential equation is not exact, but belongs to one of the types of Arts. 62-68, it may be integrated, as already seen, by a quadrature. In case the given differential equation does not belong to one of the types established, Arts. 62-68, the method of Art. 71 should be employed to find the G1 of which the equation admits. We give below a table of types of the most important of the simpler G1 in the plane, with the corresponding type of invariant differential equation. The reader will do well to re-establish for himself those types given below which were not established, Arts. 62-68. Group of One Parameter. Type of Invariant Differential Equation. WW=% d)y=F(y). (2) Ufm%. (2)y = F(#). (r)UHYA%-8)/-F- + It is seen that (3) includes types (1) and (2). wnr-g (4)y=. Equations of the form (a'.v + b'y + c')dy-ax + by + c)dx = 0 may usually be brought to the homogeneous form. See Art. 65. (6) tyS-3, +.r (6) = F(,2+y2). (7) Uf=x%-y%. mA(wdy-Uxy)ydx=0. (8) Uf-. (8)y-K)y-()=0. The form y'-f(x).y-jr(x)yn = 0 may be reduced to this one. See Art. 69. (42) Find the curve whose subnormal is constant, and equal to 2a. (43) Find the curve in which the angle between the radius vector and the ta.
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