associated homogeneous linear differential equations Can someone please explain how  associated homogeneous linear differential equations work with an example?

Lets say you have the linear differential equation
$y'' + y =3x$;
The associated homogeneous equation is
$y'' + y = 0$
The set of the solutions to the homogeneous equation is {$\alpha \cos  +\beta \sin ; \alpha, \beta \in \Bbb R$}.
One particular solution to the innitial equation is 3x.
Thus theset of solutions to the initial equation is {$\alpha \cos x +\beta \sin x + 3x; \alpha, \beta \in \Bbb R$}.
