Find the inverse Laplace of e − 2 s − 3 e − 4 s s + 2 .

Name: Find the inverse Laplace of e − 2 s − 3 e − 4 s s + 2 .
Uploaded: 2022-02-23T17:22:57+00:00
Description: Find the inverse Laplace of e − 2 s − 3 e − 4 s s + 2 .

Question

Find the inverse Laplace of                     e                  −          2          s                    −      3              e                  −          4          s                            s      +      2        .

Allvin03 · Accepted Answer

Using the Laplace&#039;s Frequency Translation Property:      L          −      1        {      e          −      a      s        }  =  H  (  t  −  a  )  f  (  t  −  a  )with H(t) the heavyside function, or step function. This function points out that everything before the point t=a is zero.So  F  (  s  )      e          −      2      s        →  H  (  t  −  2  )  f  (  t  −  2  )  F  (  s  )      e          −      4      s        →  H  (  t  −  4  )  f  (  t  −  4  )And remembering that:  L  {      1          s      +      2        −      3          (      s      +      2      )        }  =      e          −      2      t        −  3      e          −      2      t      We have just to put them together, so the solution is:  H  (  t  −  2  )      e          −      2      (      t      −      2      )        −  3  H  (  t  −  2  )      e          −      2      (      t      −      4      )

Dangelo Rosario · Accepted Answer

Using the Properties of Laplace transform,

as L {e^{b t}} = \frac{1}{s - b} ⟹ L^{- 1} (\frac{1}{s - b}) = e^{b t} ⟹ L^{- 1} (\frac{1}{s + 2}) = e^{- 2 t}

Let

F (S) = \frac{1}{s + 2}

and

f (t) = e^{- 2 t}

Now, L^{- 1} (e^{- a s} F (s)) = f (t - a) u (t - a)

where u(t) is the Heaviside step function

Find the inverse Laplace of (e^(-2s)−3e^(-4s))/(s+2)

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