Digital Library of Mathematical Functions
About the Project
NIST
4 Elementary FunctionsHyperbolic Functions

§4.34 Derivatives and Differential Equations

4.34.1 zsinhz =coshz,
4.34.2 zcoshz =sinhz,
4.34.3 ztanhz =sech2z,
4.34.4 zcschz =-cschzcothz,
4.34.5 zsechz =-sechztanhz,
4.34.6 zcothz =-csch2z.

With a0, the general solutions of the differential equations

4.34.7 2wz2-a2w =0,
4.34.8 (wz)2-a2w2 =1,
4.34.9 (wz)2-a2w2 =-1,
4.34.10 wz+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).