# Initial value problem

1. Mar 19, 2012

### fufufu

1. The problem statement, all variables and given/known data

Solve IVP
y'' + 2y' + 2y = u_pi(t) + u_2pi(t)
with IC
y(0) = 0 and y'(0) = 1.

2. Relevant equations
L{f''(t)} = s^2Y(s) - sy(0) - y'(0)
u_c(t) = u(t-c) -->Laplace--> e^-cs/s

3. The attempt at a solution
y'' + 2y' + 2y = u_pi(t) + u_2pi(t)
y'' + 2y' + 2y = u(t-pi) + u(t-2pi)
L{y'' + 2y' + 2y} = e^(-pis)/s + e^(-2pis)/s

s^2Y(s) - sy(0) - y'(0) + 2[sY(s)-y(0)] +2Y(s) = e^(-pis)/s + e^(-2pis)/s
s^2Y(s) - 1+ 2sY(s) +2Y(s) = e^(-pis)/s + e^(-2pis)/s
Y(s)(s^2 + 2s +2) = e^(-pis)/s + e^(-2pis)/s + 1
Y(s)((s+1)^2) = e^(-pis)/s + e^(-2pis)/s + 1
Y(s) = e^(-pis)/(s(s+1)^2) + e^(-2pis)/(s(s+1)^2) + 1/((s+1)^2)

before going any further though, I have hunch that the laplace of RHS is incorrect.. it looks correct when i compare it to the equation im using (above) but I am wondering when or under what circumstances the denominator of e^-cs/s can change (ie, become squared or something like that)?
thanks for any help
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution