For homogeneous ODE's with constant coefficients you need intial conditions such as y(x0)=y0 and y'(x0)=y1 or some boundary values to find a particular solution because you need to find the constants c1 and c2 in the general solution equation y=c1 $e^{r1x}$$ + $ c2 $e^{r2x} $ where r1 and r2 are roots of the auxillary equation.
But for non-homogeneous ODE's you can find a particular solution without having to know any intial conditions or boundary values.
Why ?
No, you don't need initial conditions to find a particular solution of the homogeneous case. For example, $e^{r_1 x}$ is a particular solution.