Question

(D^4 +13D^2 + 36)y = 0

Answer 1

Characteristic (auxiliary) equation

$$r^4+13r^2+36=0$$

$$r^2=-4\ or\ r^2=-9$$

$$r_1=2i, r_2=-2i, r_3=3i, r_4=-3i$$

The general solution of the given differential equation is

$$y=c_1\cos(2x)+c_2\sin(2x)+c_3\cos (3x)+c_4\sin (3x)$$