boitshupoO

Answered

2020-11-05

Find the inverse laplace transform of the function

$Y(s)=\frac{{e}^{-s}}{s(2s-1)}$

Answer & Explanation

unett

Expert

2020-11-06Added 119 answers

Step 1

The Laplace transform is given$Y(s)=\frac{{e}^{-s}}{s(2s-1)}$

Convert the laplace transform into partial derivative ,

$\frac{1}{s(2s-1)}=\frac{A}{s}+\frac{B}{2s-1}$

$1=A(2s-1)+B(s)$

$1=2As-A+Bs$

$1=s(2A+B)-A$

Compare the coefficient of terms,

$A=-1$

$2A+B=0\dots 2$

Step 2 Substitute value of A in equation 2,

$2(-1)+B=0$

$B=2$

The value of A is -1 and B is 2.

The expression is written as,

$\frac{1}{s(2s-1)}=\frac{-1}{s}+\frac{2}{2s-1}$

$=\frac{-1}{s}+\frac{1}{(s-\frac{1}{2}}$

For finding the inverse Laplace transform , use theorem below.

${L}^{-1}[{e}^{-sT}F(s)]=f(t-T)u(t-T)$

Step 3

Applying the theorem , the given Laplace transform is written as,

${L}^{-1}[{e}^{-s}(\frac{-1}{s}+\frac{1}{(s-\frac{1}{2}}]=({e}^{(\frac{t}{2}-\frac{1}{2})}-1)u(t-1)$

The inverse Laplace transform of the function$Y(s)=\frac{{e}^{-s}}{s(2s-1)}$ is , $({e}^{(\frac{t}{2}-\frac{1}{2})}-1)u(t-1)$

The Laplace transform is given

Convert the laplace transform into partial derivative ,

Compare the coefficient of terms,

Step 2 Substitute value of A in equation 2,

The value of A is -1 and B is 2.

The expression is written as,

For finding the inverse Laplace transform , use theorem below.

Step 3

Applying the theorem , the given Laplace transform is written as,

The inverse Laplace transform of the function

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