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elimination of first derivative

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1: 1.13 Differential Equations
Elimination of First Derivative by Change of Dependent Variable
Elimination of First Derivative by Change of Independent Variable
2: 19.18 Derivatives and Differential Equations
§19.18 Derivatives and Differential Equations
§19.18(i) Derivatives
Let j = / z j , and 𝐞 j be an n -tuple with 1 in the j th place and 0’s elsewhere. …
19.18.6 ( x + y + z ) R F ( x , y , z ) = 1 2 x y z ,
If n = 2 , then elimination of 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). …
3: 3.5 Quadrature
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 , , in (3.5.9) are eliminated, according to the formula … With j = 2 and k = 7 , the coefficient of the derivative f ( 16 ) ( ξ ) in (3.5.13) is found to be ( 0.14 ) × 10 13 . … For the latter a = 1 , b = 1 , and the nodes x k are the extrema of the Chebyshev polynomial T n ( x ) 3.11(ii) and §18.3). …
4: 19.25 Relations to Other Functions
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 ) . …