![{\displaystyle u_{\mu }a=A_{\mu }U_{\mu }+B_{\mu }V_{\mu },}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4194fed58aff7a6e5297e82bf63b973133b0541c)
et
![{\displaystyle {\frac {\operatorname {d} u_{\mu }}{\operatorname {d} x}}=A_{\mu }{\frac {\operatorname {d} U_{\mu }}{\operatorname {d} x}}+B_{\mu }{\frac {\operatorname {d} V_{\mu }}{\operatorname {d} x}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab2c68513a7397e8f499f51f39d182ba82a4217)
![{\displaystyle u_{\mu +1}=A_{\mu +1}U_{\mu +1}+B_{\mu +1}V_{\mu +1},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2ec57a95b0acfcd4f8627ea391db22a40d19051)
![{\displaystyle {\frac {\operatorname {d} u_{\mu +1}}{\operatorname {d} x}}=A_{\mu +1}{\frac {\operatorname {d} U_{\mu +1}}{\operatorname {d} x}}+B_{\mu +1}{\frac {\operatorname {d} V_{\mu +1}}{\operatorname {d} x}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/816062fd01648b7f5892df6caa444c46875b90e2)
Soit fait,
et nommons
![{\displaystyle \left(U_{\mu }\right),\ \left(V_{\mu }\right),\ \left({\frac {\operatorname {d} U_{\mu }}{\operatorname {d} x}}\right),\ \left({\frac {\operatorname {d} V_{\mu }}{\operatorname {d} x}}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/097284dd014b3ea3c6298b398e1adaa41fa795db)
![{\displaystyle \left(U_{\mu +1}\right),\ \left(V_{\mu +1}\right),\ \left({\frac {\operatorname {d} U_{\mu +1}}{\operatorname {d} x}}\right),\ \left({\frac {\operatorname {d} V_{\mu +1}}{\operatorname {d} x}}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/594979daf2603d7b2b96574c71b92ad74407f214)
les valeurs correspondantes des quantités comprises entre les parenthèses. Les équations de condition seront
![{\displaystyle A_{\mu }(U_{\mu })+B_{\mu }(V_{\mu })=A_{\mu +1}(U_{\mu +1})+B_{\mu +1}(V_{\mu +1}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/daffabe90c1e4027fca4377b4320414227bce778)
![{\displaystyle A_{\mu }\left({\frac {\operatorname {d} U_{\mu }}{\operatorname {d} x}}\right)+B_{\mu }\left({\frac {\operatorname {d} V_{\mu }}{\operatorname {d} x}}\right)=A_{\mu +1}\left({\frac {\operatorname {d} U_{\mu +1}}{\operatorname {d} x}}\right)+B_{\mu +1}\left({\frac {\operatorname {d} V_{\mu +1}}{\operatorname {d} x}}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a50009632277ae43a15f476b44c47d1c4f010108)
On tire de là les valeurs de
en
Le dénominateur commun de ces valeurs est
![{\displaystyle \left(U_{\mu +1}\right)\left({\frac {\operatorname {d} V_{\mu +1}}{\operatorname {d} x}}\right)-\left(V_{\mu +1}\right)\left({\frac {\operatorname {d} U_{\mu +1}}{\operatorname {d} x}}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e1eefd4ac6fa9e7753c983e28dd39d1d4ba4c48)
Or, il est remarquable que ce dénominateur est constamment égal à
Cela résulte des équations