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Physics LibreTexts

2.8: Multivariable First Integral

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Following and generalizing the one-variable derivation, multiplying the above equations one by one by the corresponding yi=dyi/dx we have the n equations

f(y,y)yidyidxddx(f(y,y)yi)yi=0

Since f doesn’t depend explicitly on x, we have

dfdx=ni=1(fyidyidx+fyidyidx)

and just as for the one-variable case, these equations give

ddx(ni=1yifyif)=0

and the (important!) first integral ni=1yifyif=constant.


This page titled 2.8: Multivariable First Integral is shared under a not declared license and was authored, remixed, and/or curated by Michael Fowler.

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