Question

Check by differentiation that y 2 cos 3 + 3 sin 3t is a solution to +9y-0 by finding the terms in the sum: y" -18 cos 31-27 sin 31 18 cos 31+27 sin 31 9y So y +9y=0

Answer 1

the expression simplifies to zero. Therefore, y = 2cos(3t) + 3sin(3t) is solution to the differential equation y'' + 9y = 0.

Explanation:

First derivative:

y' = -6sin(3t) + 9cos(3t)

Second derivative:

y'' = -18cos(3t) - 27sin(3t)

Now we substitute these derivatives into the differential equation:

y'' + 9y = (-18cos(3t) - 27sin(3t)) + 9(2cos(3t) + 3sin(3t))

= -18cos(3t) - 27sin(3t) + 18cos(3t) + 27sin(3t)

= 0