laplace 9y 6y y 0 y03 y01SolutionWe begin by applying the

laplace 9y\'\' -6y\' +y = 0 ; y(0)=3, y\'(0)=1

Solution

We begin by applying the Laplace transform to both sides. By linearity of the Laplace transform, we have

9L{y\'\'} - 6 L{y\'} + L{y} = 0

9(s²L{y} - 2s - 1) - 6 (sL{y} - 2) + L{y} =0

9s²L{y} - 18s - 9 -6sL{y} +12 + L{y} =0

L{y}(9s² - 6s + 1) = 18s - 3

L{y}= 18s - 3/(9s² - 6s + 1)

To find y, we need to take the Inverse Laplace Transform of the right hand side.

18s -3/(3s-1)^2 = A/(3s-1)^2

A = 18s -3

Let s= 1/3 then A = 3

L{y} = 3/(3s-1)^2

y =  L^-13/(3s-1)^2