By George Chrystal
This Elibron Classics e-book is a facsimile reprint of a 1904 version through Adam and Charles Black, London.
Read Online or Download Algebra: An Elementary Text-Book for the Higher Classes of Secondary Schools and for Colleges. Part 1 PDF
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Additional info for Algebra: An Elementary Text-Book for the Higher Classes of Secondary Schools and for Colleges. Part 1
Solution. With y ′ = z( x ), we have y ′′ = z′( x ) and the equation becomes z′ − z / x = x. This is the linear nonhomogeneous first-order equation. Its solution is the sum of the general solution of the homogeneous equation and the particular solution of the non-homogeneous equation. The homogeneous equation is z′ − zx = 0 , or dzz = dxx . Its general solution is z = C1x. The particular solution of the nonhomogeneous equation can be found using the method of the variation of a parameter: z = C1( x )x.
T − T1) Integration gives the general solution: T − T1 = Ce − kt . Substituting the initial condition T (0) = T0 , we find C = T0 − T1. Substituting this constant in the general solution gives the particular solution: T (t ) = T1 + e − kt (T0 − T1). 53) Now suppose that the object is heated up. 52) has the positive sign. 53) where T0 – T1 < 0. 51) also describes diffusion into a surrounding medium, where concentration of the diffusing substance is constant. 28: Radioactive decay. In radioactive decay, the mass m of a radioactive substance decreases with time, and the speed of decay, dm(t )/ dt , is proportional to a mass that has not yet disintegrated.
With the new variable z = 1+ z 2z C |z 2 −1| 2 gives z + xz′ = 1+ z 2z 2 y x , we have y ′ = z + xz′ . Equating . Next, separate variables, = 2zdz 1− z 2 , and inte- , C > 0. Thus, x = 1−Cz 2 , where C is arbitrary constant. Substitution z = 2 y = x 2 − Cx is the general solution. indb 19 dx x y x gives x = Cx 2 x2 − y 2 . 12 y ′ = y / x + tan(y / x ) First, introduce new variable z: y = xz, y ′ = z + xz′ . Then the equation becomes x dz dx + z = z + tan z. z dz ln|sin z | = ln| x | + ln C , C>0.
Algebra: An Elementary Text-Book for the Higher Classes of Secondary Schools and for Colleges. Part 1 by George Chrystal