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**[SOLVED] Laplace Transform question**

**1. Homework Statement**

Use the Laplace transform to solve the given IVp

[tex]

{\frac {d^{2}}{d{t}^{2}}}y \left( t \right) +2\,{\frac {d}{dt}}y

\left( t \right) +5\,y \left( t \right) ={e^{-t}}\sin \left( 2\,t

\right)

[/tex]

[tex]y(0)=1, y'(0)=-1[/tex]

**2. Homework Equations**

[tex]L(f) = \int^{\infty}_{0}e^{-st}f(t)dt[/tex]

[tex]

L \left( {e^{-t}}\sin \left( 2\,t \right) \right) =2\, \left(

\left( s+1 \right) ^{2}+4 \right) ^{-1}

[/tex]

[tex] y'' = Ys^{2}-y(0)s-y'(0)[/tex]

[tex] y' = Ys - y(0)[/tex]

**3. The Attempt at a Solution**

I first converted the problem into frequency space like so:

[tex] Y(s^{2}+2s+5) = 2\, \left(

\left( s+1 \right) ^{2}+4 \right) ^{-1} +2s+3[/tex]

[tex]Y={\frac {2\,{s}^{3}+7\,{s}^{2}+16\,s+17}{ \left( {s}^{2}+2\,s+5

\right) ^{2}}}[/tex]

Then I used Partial Fractions:

[tex]

Y={\frac {As+B}{{s}^{2}+2\,s+5}}+{\frac {Cs+D}{ \left( {s}^{2}+2\,s+5

\right) ^{2}}}

[/tex]

Solving the 4 equations that result I got the following values for the constants:

[tex] A = 2[/tex]

[tex] B = 3[/tex]

[tex] C = 0[/tex]

[tex] D = 2[/tex]

Putting these constants back into my initial partial fractions equation i get:

[tex]Y={\frac {2\,s+3}{{s}^{2}+2\,s+5}}+2\, \left( {s}^{2}+2\,s+5 \right) ^

{-2}[/tex]

[tex]

Y=2\,{\frac {s+1}{ \left( s+1 \right) ^{2}+4}}+ \left( \left( s+1

\right) ^{2}+4 \right) ^{-1}+2\, \left( {s}^{2}+2\,s+5 \right) ^{-2}

[/tex]

The first of these two I can find the inverse Laplaces for, but the last I cant figure out what to do with. This is where I am stuck at:

[tex]

y=2\,{e^{-t}}\cos \left( 2\,t \right) +1/2\,{e^{-t}}\sin \left( 2\,t

\right) +\mbox {{\tt `\#msup(mi("L"),mo("`}}L^{-1}\mbox {{\tt `"))`}}

\left( 2\, \left( {s}^{2}+2\,s+5 \right) ^{-2} \right) [/tex]

I cannot figure out what to do with the last term -- I dont know how to put it into a form which i can find the inverse laplace transform for. If someone could please tell me what to do with that I would appreciate it greatly.