En suivant ce procédé, on aura généralement
![{\displaystyle (a)\qquad \qquad \Delta ^{n}u=\alpha ^{n}{\frac {d^{n}u}{dx^{n}}}+q\alpha ^{n+1}{\frac {d^{n+1}u}{dx^{n+1}}}+q'\alpha ^{n+2}{\frac {d^{n+2}u}{dx^{n+2}}}+\ldots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27ffa67934dd5305da208626570df52ac089fc26)
L’équation
donne
![{\displaystyle u=\alpha \sum {\frac {du}{dx}}+h\alpha ^{2}\sum {\frac {d^{2}u}{dx^{2}}}+\ldots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9eca04c8023da93a57b4407c60161b04c1c8031b)
Soit
on aura
![{\displaystyle \sum z={\frac {1}{\alpha }}\int zdx-h\alpha \sum {\frac {dz}{dx}}-\ldots \,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15cb82fba37c95310443adb2d1525ddaaa2be55a)
on aura pareillement
![{\displaystyle \sum {\frac {\partial z}{\partial x}}={\frac {1}{\alpha }}z-h\alpha \sum {\frac {d^{2}z}{dx^{2}}}-\ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/350ee0fcef0afcda1fee5a4ca38929841b732356)
et ainsi de suite ; d’où je tire
![{\displaystyle \sum z={\frac {1}{\alpha }}\int zdx-mz-\alpha m'{\frac {dz}{dx}}-\ldots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d2212855e578c628fc62b8f52d6e3a84881bf83)
On aura pareillement
![{\displaystyle \sum ^{2}z={\frac {1}{\alpha }}\sum \int zdx-m\sum z-\alpha m'\sum {\frac {dz}{dx}}-\ldots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eafbc4da252c230a44d8a020a63dc830dfd6f819)
Or
![{\displaystyle \sum \int zdx={\frac {1}{\alpha }}\int ^{2}zdx^{2}-m\int zdx-\ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/159a3d59d0821ab2db6ced07fa6582c00014ad1c)
d’où l’on voit que
a une expression de cette forme
![{\displaystyle \sum ^{2}z={\frac {1}{\alpha ^{2}}}\int ^{2}zdx^{2}+{\frac {\mathrm {K} }{\alpha }}\int zdx+\mathrm {K} 'z+\mathrm {K} ''\alpha {\frac {dz}{dx}}+\ldots \,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19530152c0126bbf1c698c6b16ab47fe30fadc1d)
en suivant ce procédé, on conclura généralement
![{\displaystyle (b)\quad \sum ^{n}u={\frac {1}{\alpha ^{n}}}\int ^{n}udx^{n}+{\frac {r}{\alpha ^{n-1}}}\int ^{n-1}udx^{n-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a09be6bdb5889019a899ecc0f16732449455a45)
![{\displaystyle +{\frac {r'}{\alpha ^{n-2}}}\int ^{n-2}udx^{n-2}+\ldots \,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fa04f4d2b82f09c0fea6ea289727ac150c91455)
étant des coefficients constants et indépendants de ![{\displaystyle \alpha .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/794850adc0db51d11a6d8cfa857538183424909c)