I tried to change it to $y" + y = (14\sin x-28 \sin^3 x)$. The complementary solution is $C_1\cos x+C_2\sin x$ and the particular solution to $y" + y = 14 \sin x$ is $-7\sin x$. How do you find the particular solution to $y" + y = -28\sin^3 x$ ? What would be your guess?
Ans: $$c_1 \sin(x\sqrt2)+c_2\cos(s\sqrt2)-7\sin x-\sin(3x)$$