Question

Question 3 Unsaved

Use Laplace Transforms to solve the initial value problem.

y''+8y'+15y=0, y(0)=0, y'(0)=1

Answer 1

Explanation:

DE given is

y''+8y'+15y=0, y(0)=0, y'(0)=1

Take Laplace on the DE

We get

`s^2 Y(s) -sY(0) -y'(0)+8(sY(s)-y(0))+15Y(s) =0\\s^2 Y(s) -s(0) -1+8(sY(s)-0)+15Y(s) =0\\Y(s)(s^2+8s+15)-1=0\\Y(s) = (1)/(s^2+8s+15)`

Simplify to get

Y(s) =
`(1)/(2)[ {(1)/(s+3) -(1)/(s+5) }]`

Take inverse

`y(t) = (e^(-3t) -e^(-5t))/(2)`