In this problem y c1 cos 3x c2 sin 3x is a twoparameter fa

In this problem, y = c_1 cos 3x + c_2 sin 3x is a two-parameter family of solutions of the second-order DE y\" + 9y = 0. If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions. (If not possible, enter NOT.) y(0) = 0, y\'(pi/6) = 0 y =

Solution

y\"+9y=0

Auxiliary eq=0

is D^2+9=0

(D+3i)(D-3i)=0

D=3I,-3i

Solution is y(x)=c1 cos3x+c2sin3x

y\'(x)=-c1sinx+c2cosx

y\'(0)=c2=0

y\'(pi/6)=-c1(1/2)=0

Implies that c2=0

Hence y=0 (0sin3x+0cos3x)