Answer
$\{\cos 3x, \sin 3x\}$ is a basis for the solution space.
Work Step by Step
Solve the characteristic equation for the differential equation. $$r^2+9=0$$
Factor and solve for the roots. $$(r^2+9)=0 $$ $$r=3 i, -3i$$
The general equation is equal to $y=C_1 cos 3x+C_2 sin 3x$
Therefore, $\{\cos 3x, \sin 3x\}$ is a basis for the solution space.