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Showing content from https://dlmf.nist.gov/4.34 below:

§4.34 Derivatives and Differential Equations ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions

§4.34 Derivatives and Differential Equations 4.34.1 d d z ⁡ sinh ⁡ z = cosh ⁡ z , 4.34.2 d d z ⁡ cosh ⁡ z = sinh ⁡ z , 4.34.3 d d z ⁡ tanh ⁡ z = sech2 ⁡ z , 4.34.4 d d z ⁡ csch ⁡ z = − csch ⁡ z ⁢ coth ⁡ z , 4.34.5 d d z ⁡ sech ⁡ z = − sech ⁡ z ⁢ tanh ⁡ z , 4.34.6 d d z ⁡ coth ⁡ z = − csch2 ⁡ z .

With a ≠ 0 , the general solutions of the differential equations

4.34.7 d2 w d z 2 − a2 ⁢ w = 0 , 4.34.8 ( d w d z ) 2 − a2 ⁢ w2 = 1 , 4.34.9 ( d w d z ) 2 − a2 ⁢ w2 = − 1 , 4.34.10 d w d z + a2 ⁢ w2 = 1 ,

are respectively

4.34.11 w = A ⁢ cosh ⁡ ( a ⁢ z ) + B ⁢ sinh ⁡ ( a ⁢ z ) , 4.34.12 w = ( 1 / a ) ⁢ sinh ⁡ ( a ⁢ z + c ) , 4.34.13 w = ( 1 / a ) ⁢ cosh ⁡ ( a ⁢ z + c ) , 4.34.14 w = ( 1 / a ) ⁢ coth ⁡ ( a ⁢ z + c ) ,

where A , B , c are arbitrary constants.

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

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