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4 Elementary FunctionsTrigonometric Functions

§4.20 Derivatives and Differential Equations

4.20.1 ddzsinz =cosz,
4.20.2 ddzcosz =-sinz,
4.20.3 ddztanz =sec2z,
4.20.4 ddzcscz =-csczcotz,
4.20.5 ddzsecz =secztanz,
4.20.6 ddzcotz =-csc2z,
4.20.7 dndznsinz =sin(z+12nπ),
4.20.8 dndzncosz =cos(z+12nπ).

With a0, the general solutions of the differential equations

4.20.9 d2wdz2+a2w =0,
4.20.10 (dwdz)2+a2w2 =1,
4.20.11 dwdz-a2w2 =1,

are respectively

4.20.12 w =Acos(az)+Bsin(az),
4.20.13 w =(1/a)sin(az+c),
4.20.14 w =(1/a)tan(az+c),

where A,B,c are arbitrary constants.

For other differential equations see Kamke (1977, pp. 355–358 and 396–400).