An integral transform is defined as

$F(w)={\displaystyle \underset{0}{\overset{L}{}}}f(x)sin(\frac{n}{L}x)dx$

where $n=1,2,3,$dots$,$ and $L$ is a constant for the $x$ domain. Apply the integration by parts to prove that this transform to $\frac{d^{2}f}{d{x}^2}($i$.$e$.,{\displaystyle \underset{0}{\overset{L}{}}}\frac{d^{2}f}{d{x}^2}sin(\frac{n}{L}x)dx)$ requires two boundary conditions: $f=A$ at $x=0,$ and $f=B$ at $x=L$ with A and $B$ being constants. Explain whether this transform can be used to solve the ordinary differential equation $\frac{d^{2}f}{d{x}^2}f=0$ with $f=1$ at $x=0$ and $d\frac{f}{d}x=1$ at $x=L.(25$ points$)$