1. Integrating factor of a 1-form

Giving a Frobenius integrable 1-form $\omega$ necessarily there exist a smooth function $F$ and a non vanishing smooth function $\mu$ such that $$

dF=\mu \omega.

$$

En tal caso, $\mu$ recibe el nombre de integrating factor y $F$ es una first integral of the Pfaffian system $\mathcal S(\omega)$. Tiene relación inversa con las symmetrising factors.

2. Integrating factor of an ODE

Given an ODE $\Delta$, we call integrating factor of $\Delta$ to any non vanishing smooth function $\mu\in J^m$ (the jet bundle) such that

$$ D_x(F)=\mu\Delta $$

for a smooth function $F$ called first integral.

To see the relation between integrating factor of an ODE and the integrating factor of the associated 1-form go to the note first integral#3 b First integal of an m th-order ODE.

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es


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