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4 Elementary FunctionsHyperbolic Functions

§4.34 Derivatives and Differential Equations

4.34.1 ddzsinhz =coshz,
4.34.2 ddzcoshz =sinhz,
4.34.3 ddztanhz =sech2z,
4.34.4 ddzcschz =cschzcothz,
4.34.5 ddzsechz =sechztanhz,
4.34.6 ddzcothz =csch2z.

With a0, the general solutions of the differential equations

4.34.7 d2wdz2a2w =0,
4.34.8 (dwdz)2a2w2 =1,
4.34.9 (dwdz)2a2w2 =1,
4.34.10 dwdz+a2w2 =1,

are respectively

4.34.11 w =Acosh(az)+Bsinh(az),
4.34.12 w =(1/a)sinh(az+c),
4.34.13 w =(1/a)cosh(az+c),
4.34.14 w =(1/a)coth(az+c),

where A,B,c are arbitrary constants.

For other differential equations see Kamke (1977, pp. 289–400).