11
INDÉTERMINÉES.
![{\displaystyle Const.=\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'}}\right)_{0}-\left({\tfrac {\operatorname {d} V}{\operatorname {d} y''}}\right)_{0}'+\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{0}''-\ldots \right]Y_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5dda83d450dadac6f2dc312c567fd958c57912c)
![{\displaystyle +\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} y''}}\right)_{0}-\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{0}'+\ldots \right]Y_{0}'+\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{0}-\ldots \right]Y_{0}''+\ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46550b37251c9703cd368814af8e38c829039dc4)
![{\displaystyle Const.=\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'}}\right)_{1}-\left({\tfrac {\operatorname {d} V}{\operatorname {d} y''}}\right)_{1}'+\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{1}''-\ldots \right]Y_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6893cdc74b5932d3d694843f337756fe1d81b912)
![{\displaystyle +\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} y''}}\right)_{1}-\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{1}'+\ldots \right]Y_{1}'+\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{1}-\ldots \right]Y_{1}''+\ldots \,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04ac76a39e47c4501eae8093fac372701b0e3c10)
d’où en retranchant,
![{\displaystyle =\left\{{\begin{aligned}&\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'}}\right)_{1}-\left({\tfrac {\operatorname {d} V}{\operatorname {d} y''}}\right)_{1}'+\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{1}''-\ldots \right]Y_{1}\\\\&+\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} y''}}\right)_{1}-\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{1}'+\ldots \right]Y_{1}'+\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{1}-\ldots \right]Y_{1}''+\ldots \\\\&-\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'}}\right)_{0}-\left({\tfrac {\operatorname {d} V}{\operatorname {d} y''}}\right)_{0}'+\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{0}''-\ldots \right]Y_{0}\\\\&-\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} y''}}\right)_{0}-\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{0}'+\ldots \right]Y_{0}'-\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{0}-\ldots \right]Y_{0}''-\ldots \end{aligned}}\right\}(\mathrm {V} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abc468bd0e1ef9c2fa4f39b85c8a8aded5747624)
équations que nous appellerons à l’avenir équation aux limites, et qui, comme l’on voit, ne renferme plus, outre les valeurs encore indéterminées de
aux deux extrémités de l’intégrale, que les deux limites
et les constantes introduites par l’intégration de l’équation (III).
10. Cela posé, si aucune condition particulière n’a été prescrite relativement aux limites, les fonctions
![{\displaystyle Y_{0},Y'_{0},Y''_{0},\ldots Y_{1},Y'_{1},Y''_{1},\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/70eab2081fdc7e5804ccbd3a237bb69da36a791f)
devront conserver l’indépendance la plus absolue. L’équation (V) ne pourra donc alors subsister qu’autant que les coefficiens de ces diverses fonctions seront séparément nuls ; cette équation (V) se partagera donc dans les suivantes :
![{\displaystyle {\begin{aligned}&=\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'}}\right)_{0}-\left({\tfrac {\operatorname {d} V}{\operatorname {d} y''}}\right)_{0}'+\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{0}''-\ldots ,\\\\&=\left({\tfrac {\operatorname {d} V}{\operatorname {d} y''}}\right)_{0}-\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{0}'+\ldots ,\\\\&=\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{0}-\ldots ,\\\\&=\ldots \ldots \ldots \end{aligned}}\left.{\begin{aligned}&0=\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'}}\right)_{1}-\left({\tfrac {\operatorname {d} V}{\operatorname {d} y''}}\right)_{1}'+\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{1}''-\ldots ,\\\\&0=\left({\tfrac {\operatorname {d} V}{\operatorname {d} y''}}\right)_{1}-\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{1}'+\ldots ,\\\\&0=\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{1}-\ldots ,\\\\&0=\ldots \ldots \ldots \end{aligned}}\right\}(\mathrm {VI} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b4af7a03039b35e2e004b52e4df7c2ba611111b)