§4.34 Derivatives and Differential Equations

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

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

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

are respectively

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

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

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