1- find the inverse Laplace transforms of the following functions: A- f1(s)= 50(

Question

1- find the inverse Laplace transforms of the following functions:
A- f1(s)= 50(s+10)/((s+5)(s+20))

b- f2(s)= (2s^2)((s+200)(s+500))

2- find the inverse Laplace transforms of the following functions and sketch their waveforms for >0 :

a- f1(s)= ((s+))/(s(s^2+^2))

b- f2(s)=(s(s+))/((s^2+^2))

3-find the inverse Laplace transforms of the following functions :

a- f1(s)= ^2/(s^2(s+))

b- f2(s)=^2/(s(s+)^2)

4-find the inverse Laplace transforms of the following functions :

a-f1(s)= 600/((s+10)(s+20)(s+40))

b- f2(s)= (6(s+10))/((s+5)(s+20))

5-find the inverse Laplace transforms of the following functions :

a- f1(s)= 16s/((s+2)(s^2+12s+13))

b- f2(s)= (4(s^2+4))/(s(s^2+16))

Explanation / Answer

1- find the inverse Laplace transforms of the following functions:
A- f1(s)= 50(s+10)/((s+5)(s+20))

=50*(2/(3 (s+20))+1/(3 (s+5)))

now taking inverse laplace we get

= 50/3 e^(-20 t) (e^(15 t)+2)

b- f2(s)= (2s^2)/((s+200)(s+500))

= 2+800/(3 (200+s))-5000/(3 (500+s))

inverse laplace is

= 2 (delta(t)-(2500 e^(-500 t))/3+(400 e^(-200 t))/3)

2- find the inverse Laplace transforms of the following functions and sketch their waveforms for >0 :

= sin(beta t)-cos(beta t)+1

b- f2(s)=(s(s+))/((s^2+^2))

3-find the inverse Laplace transforms of the following functions :

= alpha^2 (-1/alpha^2+e^(alpha (-t))/alpha^2+t/alpha)

b- f2(s)=^2/(s(s+)^2)

4-find the inverse Laplace transforms of the following functions :

= e^(-40 t) (e^(10 t)-1)^2 (2 e^(10 t)+1)

b- f2(s)= (6(s+10))/((s+5)(s+20))

5-find the inverse Laplace transforms of the following functions :

= 16 ((2 e^(-2 t))/7-(-e^((-6-sqrt(23)) t)+2 sqrt(23) e^((-6-sqrt(23)) t)+e^((sqrt(23)-6) t)+2 sqrt(23) e^((sqrt(23)-6) t))/(14 sqrt(23)))

b- f2(s)= (4(s^2+4))/(s(s^2+16))

= 4 (3/4 cos(4 t)+1/4)

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