# elimination of first derivative

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##### 1: 1.13 Differential Equations
###### Elimination of FirstDerivative by Change of Independent Variable
1.13.17 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\eta}^{2}}+g(z)\exp\left(2\int f(z)\,% \mathrm{d}z\right)w=0.$
##### 2: 19.18 Derivatives and Differential Equations
###### §19.18(i) Derivatives
Let $\partial_{j}=\ifrac{\partial}{\partial z_{j}}$, and $\mathbf{e}_{j}$ be an $n$-tuple with 1 in the $j$th place and 0’s elsewhere. …
19.18.6 $\left(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}+\frac{\partial}{% \partial z}\right)R_{F}\left(x,y,z\right)=\frac{-1}{2\sqrt{x}\sqrt{y}\sqrt{z}},$
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). …
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)). … 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 $c_{1}h^{2},c_{2}h^{4},\dots$, in (3.5.9) are eliminated, according to the formula … With $j=2$ and $k=7$, the coefficient of the derivative $f^{(16)}(\xi)$ in (3.5.13) is found to be $(0.14\dots)\times 10^{-13}$. … 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). …
19.25.6 $\frac{\partial F\left(\phi,k\right)}{\partial k}=\tfrac{1}{3}kR_{D}\left(c-1,c% ,c-k^{2}\right).$
19.25.12 $\frac{\partial E\left(\phi,k\right)}{\partial k}=-\tfrac{1}{3}kR_{D}\left(c-1,% c-k^{2},c\right).$
19.25.17 $F\left(\phi,k\right)=R_{F}\left(x,y,z\right),$
19.25.24 $(z-x)^{1/2}R_{F}\left(x,y,z\right)=F\left(\phi,k\right),$
Inversions of 12 elliptic integrals of the first kind, producing the 12 Jacobian elliptic functions, are combined and simplified by using the properties of $R_{F}\left(x,y,z\right)$. …