2.8: Multivariable First Integral
( \newcommand{\kernel}{\mathrm{null}\,}\)
Following and generalizing the one-variable derivation, multiplying the above equations one by one by the corresponding y′i=dyi/dx we have the n equations
∂f(→y,→y′)∂yidyidx−ddx(∂f(→y,→y′)∂y′i)y′i=0
Since f doesn’t depend explicitly on x, we have
dfdx=n∑i=1(∂f∂yidyidx+∂f∂y′idy′idx)
and just as for the one-variable case, these equations give
ddx(n∑i=1y′i∂f∂y′i−f)=0
and the (important!) first integral ∑ni=1y′i∂f∂y′i−f=constant.