Question #226326

find the inverse laplace transform of F (S)=10 (S+1)/S (S+2)(S+3)^3


1
Expert's answer
2021-08-16T14:45:28-0400
10(s+1)s(s+2)(s+3)3=As+Bs+2\dfrac{10(s+1)}{s(s+2)(s+3)^3}=\dfrac{A}{s}+\dfrac{B}{s+2}

+Cs+3+D(s+3)2+E(s+3)3+\dfrac{C}{s+3}+\dfrac{D}{(s+3)^2}+\dfrac{E}{(s+3)^3}

10(s+1)=A(s+2)(s+3)3+Bs(s+3)310(s+1)=A(s+2)(s+3)^3+Bs(s+3)^3

+Cs(s+2)(s+3)2+Ds(s+2)(s+3)+Cs(s+2)(s+3)^2+Ds(s+2)(s+3)

+Es(s+2)+Es(s+2)

s=0,10=A(2)(3)3=>A=527s=0,10=A(2)(3)^3=>A=\dfrac{5}{27}

s=2,10=B(2)(1)3=>B=5s=-2,-10=B(-2)(1)^3=>B=5

s=3,20=E(3)(1)=>E=203s=-3,-20=E(-3)(-1)=>E=-\dfrac{20}{3}

s4,0=A+B+C=>C=14027s^4, 0=A+B+C=>C=-\dfrac{140}{27}

s3,0=A(9+2)+9B+C(6+2)+Ds^3, 0=A(9+2)+9B+C(6+2)+D

=>D=509=>D=-\dfrac{50}{9}

10(s+1)s(s+2)(s+3)3=5271s+51s+2\dfrac{10(s+1)}{s(s+2)(s+3)^3}=\dfrac{5}{27}\cdot\dfrac{1}{s}+5\cdot\dfrac{1}{s+2}

140271s+35091(s+3)22031(s+3)3-\dfrac{140}{27}\cdot\dfrac{1}{s+3}-\dfrac{50}{9}\cdot\dfrac{1}{(s+3)^2}-\dfrac{20}{3}\cdot\dfrac{1}{(s+3)^3}


L1(10(s+1)s(s+2)(s+3)3)=527H(t)+5e2tL^{-1}(\dfrac{10(s+1)}{s(s+2)(s+3)^3})=\dfrac{5}{27}H(t)+5e^{-2t}

14027e3t509te3t103t2e3t-\dfrac{140}{27}e^{-3t}-\dfrac{50}{9}te^{-3t}-\dfrac{10}{3}t^2e^{-3t}



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