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1: 22.13 Derivatives and Differential Equations

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§22.13(ii) First-Order Differential Equations

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2: 19.4 Derivatives and Differential Equations

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${\displaystyle \frac{dK\left(k\right)}{dk}}={\displaystyle \frac{E\left(k\right)-k_{}^{\prime }{}_{}{}^{2}K\left(k\right)}{kk_{}^{\prime }{}_{}{}^2}},$

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${\displaystyle \frac{d(E\left(k\right)-k_{}^{\prime }{}_{}{}^{2}K\left(k\right))}{dk}}=kK\left(k\right),$

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${\displaystyle \frac{dE\left(k\right)}{dk}}={\displaystyle \frac{E\left(k\right)-K\left(k\right)}{k}},$

… ►If $\varphi =\pi /2$, then these two equations become hypergeometric differential equations (15.10.1) for $K\left(k\right)$ and $E\left(k\right)$. An analogous differential equation of third order for $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ is given in Byrd and Friedman (1971, 118.03).

3: 3.6 Linear Difference Equations

… ►Similar considerations apply to the first-order equation … ►or for systems of $k$ first-order inhomogeneous equations, boundary-value methods are the rule rather than the exception. …

4: 3.7 Ordinary Differential Equations

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First-Order Equations

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5: 18.38 Mathematical Applications

… ►The Dunkl operator, introduced by Dunkl (1989), is an operator associated with reflection groups or root systems which has terms involving first order partial derivatives and reflection terms. …

6: 32.10 Special Function Solutions

… ►All solutions of ${\text{P}}_{\text{II}}$–${\text{P}}_{\text{VI}}$ that are expressible in terms of special functions satisfy a first-order equation of the form …

7: 3.4 Differentiation

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First-Order

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