Question #343135

Inverse Laplace Transforms

Find L^-1 {F(s)} when F(s) is given by


5. 1/(s+1)(s+2)(s^2+2s+10)


1
Expert's answer
2022-05-29T16:07:55-0400

5.


1(s+1)(s+2)(s2+2s+10)=As+1+Bs+2\dfrac{1}{(s+1)(s+2)(s^2+2s+10)}=\dfrac{A}{s+1}+\dfrac{B}{s+2}




+Cs+Ds2+2s+10=A(s+2)(s2+2s+10)(s+1)(s+2)(s2+2s+10)+\dfrac{Cs+D}{s^2+2s+10}=\dfrac{A(s+2)(s^2+2s+10)}{(s+1)(s+2)(s^2+2s+10)}


+B(s+1)(s2+2s+10)(s+1)(s+2)(s2+2s+10)+\dfrac{B(s+1)(s^2+2s+10)}{(s+1)(s+2)(s^2+2s+10)}

+(Cs+D)(s+1)(s+2)(s+1)(s+2)(s2+2s+10)+\dfrac{(Cs+D)(s+1)(s+2)}{(s+1)(s+2)(s^2+2s+10)}

s=1:A(12+10)=1s=-1:A(1-2+10)=1

s=2:B(44+10)=1s=-2:-B(4-4+10)=1

s=0:20A+10B+2D=1s=0: 20A+10B+2D=1


s=1:39A+26B+6C+6D=1s=1:39A+26B+6C+6D=1

A=19,B=110,D=19,C=190A=\dfrac{1}{9}, B=-\dfrac{1}{10}, D=-\dfrac{1}{9}, C=-\dfrac{1}{90}


L1(1(s+1)(s+2)(s2+2s+10))L^{-1}(\dfrac{1}{(s+1)(s+2)(s^2+2s+10)})

=19L1(1s+1)110L1(1s+2)=\dfrac{1}{9}L^{-1}(\dfrac{1}{s+1})-\dfrac{1}{10}L^{-1}(\dfrac{1}{s+2})

190L1(s+1(s+1)2+32)110L1(1(s+1)2+32)-\dfrac{1}{90}L^{-1}(\dfrac{s+1}{(s+1)^2+3^2})-\dfrac{1}{10}L^{-1}(\dfrac{1}{(s+1)^2+3^2})



=19et110e2t190etcos(3t)130etsin(3t)=\dfrac{1}{9}e^{-t}-\dfrac{1}{10}e^{-2t}-\dfrac{1}{90}e^{-t}\cos(3t)-\dfrac{1}{30}e^{-t}\sin (3t)




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