Si l’expression de
renferme deux termes, en sorte que l’on ait
![{\displaystyle z=\mathrm {A\varphi (\varpi )+A'} \varphi _{1}(\varpi ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/025c53fdcbaff59d76b2957b6162bdeeb4f68806)
en substituant cette valeur de
dans l’équation différentielle, on aura les trois suivantes
![{\displaystyle {\begin{aligned}0=&{\frac {\partial \mathrm {A} }{\partial \theta }}+m\mathrm {A} ,\\0=&{\frac {\partial ^{2}\mathrm {A} }{\partial \varpi \partial \theta }}+m{\frac {\partial \mathrm {A} }{\partial \varpi }}\ +n{\frac {\partial \mathrm {A} }{\partial \theta }}\ +l\mathrm {A} \ +{\frac {\partial \mathrm {A} '}{\partial \theta }}+m\mathrm {A} ',\\0=&{\frac {\partial ^{2}\mathrm {A} '}{\partial \varpi \partial \theta }}+m{\frac {\partial \mathrm {A} '}{\partial \varpi }}+n{\frac {\partial \mathrm {A} '}{\partial \theta }}+l\mathrm {A} '\,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/558286abe444fddc7bfb07e6a3841f695ad21c29)
si l’on fait
![{\displaystyle z^{(1)}={\frac {\partial z}{\partial \theta }}+mz,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57a7f7f7be4bebf008ccdf141f8b8565713afe3d)
on aura
![{\displaystyle z^{(1)}=\left({\frac {\partial \mathrm {A} }{\partial \theta }}+m\mathrm {A} \right)\varphi (\varpi )+\left({\frac {\partial \mathrm {A} '}{\partial \theta }}+m\mathrm {A} '\right)\varphi _{1}(\varpi )=\left({\frac {\partial \mathrm {A} '}{\partial \theta }}+m\mathrm {A} '\right)\varphi _{1}(\varpi ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c31b8f0a689142c020d7487b490a7a1488b68272)
en sorte que l’expression de
ne renfermera qu’un seul terme ; présentement, l’équation
![{\displaystyle 0={\frac {\partial ^{2}z}{\partial \varpi \partial \theta }}+m{\frac {\partial z}{\partial \varpi }}+n{\frac {\partial z}{\partial \theta }}+lz}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c11532d672eb3ef2bd1c78ee4eefcb45127c3fe7)
peut être mise, comme on l’a vu, sous cette forme
![{\displaystyle 0={\frac {\partial z^{(1)}}{\partial \varpi }}+nz^{(1)}+z\left(l-{\frac {\partial m}{\partial \varpi }}-nm\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fab05514ecc49f3e22ed91e796fd8ff66d99b0f9)
Soit
![{\displaystyle \mu =l-{\frac {\partial m}{\partial \varpi }}-nm,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8eec6fc298c47f4d165f6acac20538da2b3f27e1)
on aura
![{\displaystyle 0={\frac {1}{\mu }}{\frac {\partial z^{(1)}}{\partial \varpi }}+{\frac {n}{\mu }}z^{(1)}+z\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8b7b5ab2e57d78c880ce0177c5ff1a8b316461f)
en différenciant cette équation par rapport à
on aura
![{\displaystyle 0={\frac {1}{\mu }}{\frac {\partial ^{2}z^{(1)}}{\partial \varpi \partial \theta }}-{\frac {\cfrac {\partial \mu }{\partial \theta }}{\mu ^{2}}}{\frac {\partial z^{(1)}}{\partial \varpi }}+{\frac {n}{\mu }}{\frac {\partial z^{(1)}}{\partial \theta }}+z^{(1)}\left({\frac {\cfrac {\partial n}{\partial \theta }}{\mu }}-n{\frac {\cfrac {\partial \mu }{\partial \theta }}{\mu ^{2}}}\right)+{\frac {\partial z}{\partial \theta }}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3871dd315ccd13c26f509397c5bfbc6d1f1b931d)