Working through Simmons' Differential Equations with Applications and Historical Notes and we're stuck in Case C, page 360.
Case C: If x is missing from the function $f(x,y,y')$, then Euler's equation can be integrated to: $$\frac{\partial f}{\partial y'}\,y'-f=c_1$$ This follows from the identity: $$\frac{d}{dx}\left(\frac{\partial f}{\partial y'}\,y'-f\right)=y'\left[\frac{d}{dx}\left(\frac{\partial f}{\partial y'}\right)-\frac{\partial f}{\partial y}\right]-\frac{\partial f}{\partial x}$$ The problem for us is what happens when we differentiate the left-hand side of the last result. \begin{align*}\frac{d}{dx}\left(\frac{\partial f}{\partial y'}\,y'-f\right) &=\frac{d}{dx}\left(\frac{\partial f}{\partial y'}\,y'\right)-\frac{\partial f}{\partial x}\\ &=\frac{d}{dx}\left(\frac{\partial f}{\partial y'}\right)\,y'+\frac{\partial f}{\partial y'}\,y''-\frac{\partial f}{\partial x} \end{align*}
So we're not getting his result. Any thoughts?
So, with your comment suggestion, and the fact that $f$ is a function of $y$ and $y'$, we use the chain rule to write
$$\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}\frac{dy}{dx}+\frac{\partial f}{\partial y'}\frac{dy'}{dx},$$
or equivalently,
$$\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}\,y'+\frac{\partial f}{\partial y'}\,y''$$
Now, because $f$ is a function of $y$ and $y'$ only, $\partial{f}/\partial{x}=0$ and we get:
$$\frac{\partial f}{\partial y'}\,y''=-\frac{\partial f}{\partial y}\,y'$$
Substituting this above, we get, because $\partial f/\partial x=0$,
\begin{align*}\frac{d}{dx}\left(\frac{\partial f}{\partial y'}\,y'-f\right) &=\frac{d}{dx}\left(\frac{\partial f}{\partial y'}\,y'\right)-\frac{\partial f}{\partial x}\\ &=\frac{d}{dx}\left(\frac{\partial f}{\partial y'}\right)\,y'+\frac{\partial f}{\partial y'}\,y''-\frac{\partial f}{\partial x}\\ &=\frac{d}{dx}\left(\frac{\partial f}{\partial y'}\right)\,y'-\frac{\partial f}{\partial y}\,y'-\frac{\partial f}{\partial x}\\ &=y'\left[\frac{d}{dx}\left(\frac{\partial f}{\partial y'}\right)-\frac{\partial f}{\partial y}\right] \end{align*}
Does this look correct?