this is initial value problem.. it is solved by using laplace transform
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 y"+y'+9y=0 y'(0)=0, y(0)=0.16

Note that ℒ{y"}=s²Y(s)-sy(0)-y'(0) and ℒ{y'}=sY(s)-y(0).

Also, for s>a, ℒ{eᵃᵗsinwt}=w/((s-a)²+w²) and ℒ{eᵃᵗcost}=(s-a)/((s-a)²+w²).

Apply ℒ to both sides:

s²Y-sy(0)-y'(0)+sY-y(0)+9Y=s²Y-0.16s+sY-0.16+9Y=0 from given initial conditions;





(These steps arrange the expression into a form which lends itself to converting back from Laplace Transform.)

Apply inverse ℒ to both sides:


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