# §4.34 Derivatives and Differential Equations

 4.34.1 $\displaystyle\frac{d}{dz}\mathop{\sinh\/}\nolimits z$ $\displaystyle=\mathop{\cosh\/}\nolimits z,$ 4.34.2 $\displaystyle\frac{d}{dz}\mathop{\cosh\/}\nolimits z$ $\displaystyle=\mathop{\sinh\/}\nolimits z,$ 4.34.3 $\displaystyle\frac{d}{dz}\mathop{\tanh\/}\nolimits z$ $\displaystyle={\mathop{\mathrm{sech}\/}\nolimits^{2}}z,$ 4.34.4 $\displaystyle\frac{d}{dz}\mathop{\mathrm{csch}\/}\nolimits z$ $\displaystyle=-\mathop{\mathrm{csch}\/}\nolimits z\mathop{\coth\/}\nolimits z,$ 4.34.5 $\displaystyle\frac{d}{dz}\mathop{\mathrm{sech}\/}\nolimits z$ $\displaystyle=-\mathop{\mathrm{sech}\/}\nolimits z\mathop{\tanh\/}\nolimits z,$ 4.34.6 $\displaystyle\frac{d}{dz}\mathop{\coth\/}\nolimits z$ $\displaystyle=-{\mathop{\mathrm{csch}\/}\nolimits^{2}}z.$

With $a\neq 0$, the general solutions of the differential equations

 4.34.7 $\displaystyle\frac{{d}^{2}w}{{dz}^{2}}-a^{2}w$ $\displaystyle=0,$ 4.34.8 $\displaystyle\left(\frac{dw}{dz}\right)^{2}-a^{2}w^{2}$ $\displaystyle=1,$ 4.34.9 $\displaystyle\left(\frac{dw}{dz}\right)^{2}-a^{2}w^{2}$ $\displaystyle=-1,$ 4.34.10 $\displaystyle\frac{dw}{dz}+a^{2}w^{2}$ $\displaystyle=1,$

are respectively

 4.34.11 $\displaystyle w$ $\displaystyle=A\mathop{\cosh\/}\nolimits\!\left(az\right)+B\mathop{\sinh\/}% \nolimits\!\left(az\right),$ 4.34.12 $\displaystyle w$ $\displaystyle=(1/a)\mathop{\sinh\/}\nolimits\!\left(az+c\right),$ 4.34.13 $\displaystyle w$ $\displaystyle=(1/a)\mathop{\cosh\/}\nolimits\!\left(az+c\right),$ 4.34.14 $\displaystyle w$ $\displaystyle=(1/a)\mathop{\coth\/}\nolimits\!\left(az+c\right),$

where $A,B,c$ are arbitrary constants.

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