Inverse Laplace Transform?

you need to do 2 things:

- decompose fraction into partial fractions

- do inverse laplace on each term

step1: find A and B such that we get set of partial fractions:

(s+3)/[(s+1)(s+2)] = A/(s+1) + B/(s+2)

(s+3)= A(s+2) +B(s+1)

s+3= (A+B)s +(2A+B)

now we match coefficients and we see that

1 = A+B

3=2A+B

solving those two equations we get

A=2

B=-1

so the original problem is now reduced to finding inverse laplace of

2/(s+1) -1/(s+2)

which is much simpler

step2:

use known relationship (from table of laplace transforms):

e^(-at) <---> 1/(s+a)

first term is

2* 1/(s+1)

so a=1 and inverse laplace is 2* e^(-1*t)

second term is -1* 1/(s+2)

so here a=2 and we get

-1*e^(-2*t)

solution is therefore

2e^(-t) -e^(-2t)