# elimination of first derivative

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## 4 matching pages

##### 1: 1.13 Differential Equations

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###### Elimination of First Derivative by Change of Dependent Variable

… ►###### Elimination of First Derivative by Change of Independent Variable

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1.13.17
$$\frac{{d}^{2}w}{{d\eta}^{2}}+g(z)\mathrm{exp}\left(2\int f(z)dz\right)w=0.$$

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##### 2: 19.18 Derivatives and Differential Equations

###### §19.18 Derivatives and Differential Equations

►###### §19.18(i) Derivatives

… ►Let ${\partial}_{j}=\partial /\partial {z}_{j}$, and ${\mathbf{e}}_{j}$ be an $n$-tuple with 1 in the $j$th place and 0’s elsewhere. … ►
19.18.6
$$\left(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}+\frac{\partial}{\partial z}\right){R}_{F}(x,y,z)=\frac{-1}{2\sqrt{x}\sqrt{y}\sqrt{z}},$$

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►If $n=2$, then elimination of ${\partial}_{2}v$ between (19.18.11) and (19.18.12), followed by the substitution $({b}_{1},{b}_{2},{z}_{1},{z}_{2})=(b,c-b,1-z,1)$, produces the Gauss hypergeometric equation (15.10.1).
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##### 3: 3.5 Quadrature

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►If $k$ in (3.5.4) is not arbitrarily large, and if odd-order derivatives of $f$ are known at the end points $a$ and $b$, then the composite trapezoidal rule can be improved by means of the Euler–Maclaurin formula (§2.10(i)).
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►As in Simpson’s rule, by combining the rule for $h$ with that for $h/2$, the first error term ${c}_{1}{h}^{2}$ in (3.5.9) can be eliminated.
With the Romberg scheme successive terms ${c}_{1}{h}^{2},{c}_{2}{h}^{4},\mathrm{\dots}$, in (3.5.9) are eliminated, according to the formula
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►With $j=2$ and $k=7$, the coefficient of the derivative
${f}^{(16)}(\xi )$ in (3.5.13) is found to be $(0.14\mathrm{\dots})\times {10}^{-13}$.
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►For the latter $a=-1$, $b=1$, and the nodes ${x}_{k}$ are the extrema of the Chebyshev polynomial ${T}_{n}\left(x\right)$ (§3.11(ii) and §18.3).
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##### 4: 19.25 Relations to Other Functions

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19.25.6
$$\frac{\partial F(\varphi ,k)}{\partial k}=\frac{1}{3}k{R}_{D}(c-1,c,c-{k}^{2}).$$

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19.25.12
$$\frac{\partial E(\varphi ,k)}{\partial k}=-\frac{1}{3}k{R}_{D}(c-1,c-{k}^{2},c).$$

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19.25.17
$$F(\varphi ,k)={R}_{F}(x,y,z),$$

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19.25.24
$${(z-x)}^{1/2}{R}_{F}(x,y,z)=F(\varphi ,k),$$

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►Inversions of 12 elliptic integrals of the first kind, producing the 12 Jacobian elliptic functions, are combined and simplified by using the properties of ${R}_{F}(x,y,z)$.
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