Find by three methods a Green's formula for the operator $\displaystyle B=\frac{d^4}{dx^4}$.
$\displaystyle \displaystyle\int_a^bf^{(4)}g \ dx$
Need a lot of help here, thanks.
My hint was to use integration by parts 4 times.
$\displaystyle \displaystyle f'''g-fg'''+f'g''-f''g'+\int_a^bfg^{(4)}dx$
Now, it says notice that $\displaystyle B=A^2$ where $\displaystyle A=-\left(\frac{d^2}{dx^2}\right)$.
Show that $\displaystyle f'''g-fg'''$ is a derivative [analogous to formula (4.6.1)]
4.6.1:
$\displaystyle f''g-fg''=\frac{d}{dx}(f'g-fg')$
Not sure how to do that, but I assume it isn't too difficult.