Verify if y=c*sin2x+3*cos 2x is the general solution for d^2y/dx^2+4y=0?

We'll have to determine the first derivative of the
function:

y' = 2c*cos 2x - 6*sin
2x

Now, we'll determine the second derivative of
y:

y" = -4c*sin 2x - 12*cos
2x

Now, we'll substitute y" and y into the given
differential equation to verify if it is
cancelling.

d^2y/dx^2 + 4y = (-4c*sin 2x - 12*cos 2x) +
4(c*sin 2x + 3*cos 2x)

We'll remove the
brackets:

d^2y/dx^2 + 4y = -4c*sin 2x - 12*cos 2x + 4c*sin
2x + 12*cos 2x

We notice that we can eliminate all terms
from the right side:

d^2y/dx^2 + 4y =
0

Therefore, y = c*sin 2x + 3*cos 2x
represents the general solution of the equation d^2y/dx^2 +
4y.