Follow the steps below to use the Laplace transform to solve

Follow the steps below to use the Laplace transform to solve the following initial value problem.

Follow the steps below to use the Laplace transform to solve the following initial value problem. y + {3} y\' = 0 hspace{0.5in} y(0) = 1, ; y\'(0)= 5 a) UsingY = mathcal{L} Ibrace y(t) rbrace, enter the equation you get by taking the Laplace transform of the differential equation above =0 b) Solve this for Y(s) = c) Finally, use inverse transforms to findy(t)=

Solution

a)

L{y\"} + 3L{y\'} = 0

s²Y - sy(0) - y\'(0) + 3(sY - y(0)) = 0

(s²+3s)Y - (s+8) = 0

b)

Y(s) = (s+8)/(s²+3s)

c)

Y(s) = (8/3)/s - (5/3)/(s+3)

Thus:

y(t) = 8/3 - 5/3 e^-3t