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I saw this as a step in a calculation and it was confusing. The left cancels down to:
$$\int \frac{d^2y}{dx^2}dy$$
But surely the answer is $\frac{d^2y}{dx^2}y$; instead, $\frac {d^2}{dx^2}$ is being treated as a constant and not $\frac {d^2y}{dx^2} $
. Putting in the calculation directly yields the result I would have expected.
$\endgroup$ 01 Answer
$\begingroup$For the left side you have that:$$I=\int y''y'dx= \frac 12\int(y'^2)'dx=\frac 12 \int \dfrac {d(y')^2}{dx}dx$$$$I=\frac 12 \int d(y')^2=\frac 12 (y')^2$$
Note that;$$(y'^2)'=2y'y''$$
$\endgroup$ 12
