I'll start by stating the problem text:
"A string is streched along the x-axis from $x=0$ to $x= \pi/2$ and is made to vibrate by a force proportional to - $f(x)sin( \omega t)$. The amplitude, $y(x)$ , of small vibrations is then given as the solution to: $$ y'' + y = f(x) $$ Where $y(0) = y(\pi/2) = 0 $ since the string is fixed at both ends. Find a Green function for the equation and use it to construct the solution the case that the driving force $f(x)$ is given by $$ f(x) = x, 0 < x < \pi /4$$ $$f(x)= \pi/2 - x, \pi/4 < 0 < \pi/2 $$ "
I'll also add that this exercise is part of a course that uses "Arfken, Weber & Harris, Mathematical methods for physicists" as course-book.
My work and my issues so far: When trying to solve this I immediately got confused because the previous stretched string problems I've encountered have a characteristic "wave equation" associated with them but in this case I can't find an explicit one stated. I started by trying to solve the homogeneous form of the 2nd order ODE that is stated in the problem but that got me nowhere ($y(x)=0$ only solution).
I just have no idea where to start with this problem, should I write out the differential equation for a stretched string with source term and then solve for a greens function for that one?
It's no problem that the only solution to the homogeneous problem that satisfies the boundary conditions is $y(x) = 0$; actually, that's the feature allowing you to construct a unique solution to your inhomogeneous ODE $y'' + y = f(x)$. I advice you to follow Wikipedia's expansion on Green's functions, both for the existence theorem and for the construction of the Green's function.