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ordinary differential equations; linear differential equations; transformations; functional equations
The paper describes the general form of an ordinary differential equation of an order $n+1$ $(n\ge 1)$ which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form \[ f\biggl (s, w_{00}v_0, \ldots , \sum _{j=0}^n w_{n j}v_j\biggr )=\sum _{j=0}^n w_{n+1 j}v_j + w_{n+1 n+1}f(x,v, v_1, \ldots , v_n), \] where $w_{n+1 0}=h(s, x, x_1, u, u_1, \ldots , u_n)$, $ w_{n+1 1}=g(s, x, x_1, \ldots , x_n, u, u_1, \ldots , u_n)$ and $w_{i j}=a_{i j}(x_1, \ldots , x_{i-j+1}, u, u_1, \ldots , u_{i-j})$ for the given functions $a_{i j}$ is solved on $\mathbb R$, $ u\ne 0.$
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[n8] V. Tryhuk: On global transformations of ordinary differential equations of the second order. Czechoslovak Math. J. 50 (125) (2000), 499–508. DOI 10.1023/A:1022825325021 | MR 1777471 | Zbl 1079.34502
[n9] V. Tryhuk: Remark to transformations of linear differential and functional-differential equations. Czechoslovak Math. J. 50 (125) (2000), 265–278. DOI 10.1023/A:1022414717364 | MR 1761386
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