# Differential Equations question.

## Homework Statement

We have this DE:
y(t)''+2y(t)'+y(t)=e^(-t)
y(0)=1
y(0)'=0

## Homework Equations

We use LT
L(y(t)'')=F(s)s^2-y(0)s-y(0)'
L(y(t)')=F(s)s-y(0)
L(y(t))=F(s)

## The Attempt at a Solution

After calculus
F(s)=(s^2+3s+3)/(s+1)^3=1/(s+1)+1/(s+1)^2+1/(s+1)^3
=> y(t)=e^(-t)(1+t+t^2/2)
Which is a good result and y(0)=1 and y(0)`=0 http://www.wolframalpha.com/input/?i=%28+e^%28-t%29+%28t^2%2F2%2Bt%2B1%29%29%27%27%2B2*%28+e^%28-t%29+%28t^2%2F2%2B+t%2B1%29%29%27%2B%28e^%28-t%29+%28t^2%2F2%2B1+%2Bt%29%29
Now, i use the formula that i was teached with residue
We have a pole of order 3 = -1
so y(t)=lim z=-1 (1/2!* (2 -- after we derivate 2 times))*e^(zt)) =e^(-t).
Which is a wrong answer and i want to ask you why???

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