find impulse and step response of Please show steps Find imp

find impulse and step response of

Please show steps

Find impulse and step response of G(s) = s + 1/s^2 + 5s + 6 please show steps

impulse response:

laplace transform of impulse response=1

g(s)=(s+1)/(s^2+5s+6)=(s+1)/[(s+2)*(s+3)]=a/(s+2)+b/(s+3)

s+1=as+3a+bs+2b

compare

a+b=1

3a+2b=1

from the above equations

a=-1,b=2

g(s)=-1/(s+2)+2/(s+3)

apply inverse laplace transform

g(t)=-1*e^-2t+2*e^-3t

step response:

laplace transform of step response=1/s

g(s)=[(s+1)/{s*(s+2)*(s+3)}]=a/s+b/(s+2)+c/(s+3)

s+1=a(s+1)(s+3)+bs*(s+3)+c*s*(s+2)

compare

a+b+c=0

5a+3b+2c=1

6a=1

a=1/6

from the above equations

a=1/6

b=1/2

c=-2/3

g(s)=(1/6)*(1/s)+(1/2)*[1/(s+2)]+(-2/3)*[1/(s+3)]

apply inverse laplace transform

g(t)=(1/6)+(1/2)*e^(-2t)+(-2/3)*e^(-3t)