Show all work for full credit Use Laplace transforms to sol

Show all work for full credit !

Use Laplace transforms to solve the initial value problem. Y\" + x + 2y = 0 x\" + x + 2y = 0 x(0) = 0, y(0) = 0 x\'(0) = 1, y\'(0) = 1

Solution

s^2Y(s)-sy(0)-y\'(0)+X(s)+2Y(s)=0
s^2X(s)-sx(0)-x\'(0)+X(s)+2Y(s)=0
s^2Y(s)-1+X(s)+2Y(s)=0
s^2X(s)-1+X(s)+2Y(s)=0
So X(s)=Y(s)
s^2X(s)+X(s)+2X(s)=1
X(s)=1/(s^2+3)
x(t)=y(t)=sin(3t)/3