Chapter 7: Q17RP (page 350)

Determine the inverse Laplace transform of the given function.
$\frac{e^{- 2 s} (4 s + 2)}{(s - 1) (s + 2)}$

Short Answer

Expert verified

The solution is $L^{- 1} \{\frac{e^{- 2 s} (4 s + 2)}{(s - 1) (s + 2)}\} = (2 e^{t - 2} + 2 e^{- 2 t + 4}) u (t - 2)$

Step by step solution

Step 1:Given Information.

The given function value in s domain is . $\frac{e^{- 2 s} (4 s + 2)}{(s - 1) (s + 2)}$

Determining the inverse Laplace transform

Make partial fraction of the given function, as:

$\frac{4 s + 2}{(s - 1) (s + 2)} = \frac{A}{s - 1} + \frac{B}{s + 2}$

On comparing the two sides of the above equation we get:

$4s+2=A(s+2)+B(s-1)$

On comparing the coefficients with $s = - 1, 2$ respectively we get

$\begin{array}{l} A = 2 \\ B = 2 \end{array}$

Thus, the partial fractions are obtained as:

$\frac{4 s + 2}{(s - 1) (s + 2)} = \frac{2}{s - 1} + \frac{2}{s + 2}$

We now have

$\begin{matrix} L^{- 1} \{\frac{e^{- 2 s} (4 s + 2)}{(s - 1) (s + 2)}\} = L^{- 1} \{\frac{2 e^{- 2 s}}{s - 1}\} + L^{- 1} \{\frac{2 e^{- 2 s}}{s + 2}\} \\ = 2 e^{t - 2} u (t - 2) + 2 e^{- 2 (t - 2)} u (t - 2) \\ = (2 e^{t - 2} + 2 e^{- 2 t + 4}) u (t - 2) \end{matrix}$

Hence, the required inverse Laplace transform is.

$L^{- 1} \{\frac{e^{- 2 s} (4 s + 2)}{(s - 1) (s + 2)}\} = (2 e^{t - 2} + 2 e^{- 2 t + 4}) u (t - 2)$

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Determine the inverse Laplace transform of the given function.
$\frac{e^{- 2 s} (4 s + 2)}{(s - 1) (s + 2)}$

Short Answer

Step by step solution

Step 1:Given Information.

Determining the inverse Laplace transform

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Determine the inverse Laplace transform of the given function.e-2s(4s+2)(s-1)(s+2)

Short Answer

Step by step solution

Step 1:Given Information.

Determining the inverse Laplace transform

One App. One Place for Learning.

Most popular questions from this chapter

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Determine the inverse Laplace transform of the given function.
$\frac{e^{- 2 s} (4 s + 2)}{(s - 1) (s + 2)}$