Question #343133

Inverse Laplace Transforms

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


3. s-1/s^2 (s+3)


1
Expert's answer
2022-05-26T11:38:12-0400
s1s2(s+3)=As+3+Bs+Cs2\dfrac{s-1}{s^2(s+3)}=\dfrac{A}{s+3}+\dfrac{B}{s}+\dfrac{C}{s^2}=As2+Bs2+3Bs+Cs+3C(s+3)(s+7)=\dfrac{As^2+Bs^2+3Bs+Cs+3C}{(s+3)(s+7)}A+B=0A+B=03B+C=13B+C=13C=13C=-1


A=49,B=49,C=13A=-\dfrac{4}{9}, B=\dfrac{4}{9}, C=-\dfrac{1}{3}




L1(1(s+3)(s+3))=49L1(1s+3)L^{-1}(\dfrac{1}{(s+3)(s+3)})=-\dfrac{4}{9}L^{-1}(\dfrac{1}{s+3})




+49L1(1s)13L1(1s2)+\dfrac{4}{9}L^{-1}(\dfrac{1}{s})-\dfrac{1}{3}L^{-1}(\dfrac{1}{s^2})




=49e3t+4913t=-\dfrac{4}{9}e^{-3t}+\dfrac{4}{9}-\dfrac{1}{3}t


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