Answer
$A=-\frac{4}{7}$
Work Step by Step
We begin by solving for the derivatives of $y$:
$y = Asin(3t)$
$\frac{dy}{dt} = Acos(3t) * (\frac{d}{dt} 3t) = 3Acos(3t)$
$\frac{d^2y}{dt^2} = -3Asin(3t) * (\frac{d}{dt}3t)=-9Asin(3t)$
We substitute these values into the equation and solve for $A$:
$\frac{d^2y}{dt^2} + 2y = 4sin(3t)$
$-9Asin(3t) + 2(Asin(3t) =4sin(3t)$
$-7Asin(3t) = 4sin(3t)$
$-7A = 4$
$A=-\frac{4}{7}$
