I'm having difficulties calculating a simple Laplace inverse

Haylee Bowen 2022-04-07 Answered
I'm having difficulties calculating a simple Laplace inverse :
S4S22S11
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Answers (1)

enchantsyseq
Answered 2022-04-08 Author has 19 answers
When Laplace transforms are rational functions, the typical line of attack is to put it into the form of partial fractions. Note that S22S11=(S1)212 has roots 1±12=1±23, so:
S4S22S11=S4(S123)(S1+23)
So, we try to write it in the form AS123+BS1+23 , with A,B constants. One way of solving for these constants is the cover-up method. We apply this here:
A=S4S1+23S=1+23=3+2343=234
B=S4S123S=123=32343=2+34
Thus, our function's Laplace transform can be written as 14(23S1+23+2+3S123)
We now recall that eat has Laplace transform 1sa , and this allows us to recover our function as:
f(t)=14((23)e(123)t+(2+3)e(1+23)t)
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