Question

Use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. (Write your answer as a function of t.) ℒ−1 s (s − 4)(s − 5)(s − 20)

Answer 1

Answer:

$L^{-1} \begin {pmatrix} \dfrac{s}{((s-20)(s-5)(s-4)} \end {pmatrix}= \dfrac{e^{4t}}{12}\begin {pmatrix}e^{16t} - 4e^t+3 \end {pmatrix}$

Step-by-step explanation:

From the given information the first step to carry out is to find the partial fraction of the given equation; i.e.

$\dfrac{s}{(s-4)(s-5)(s-20)}= \dfrac{A}{s-4}+\dfrac{B}{s-5}+\dfrac{C}{s-20}$

After finding the partial fraction; we get:

$s= s^2(A+B+C)+s(-2A-24B-25C)+20A+80B+100C$

By solving this; we get:

$A =\dfrac{1}{12}\ ; \ B = -\dfrac{1}{3} \ ; \ C= \dfrac{1}{4}$

Thus;  the above equation can be re-written as:

$\dfrac{s}{(s-20)(s-5)(s-4)}= \dfrac{\dfrac{1}{12}}{s-20}+\dfrac{-\dfrac{1}{3} }{s-5}+\dfrac{\dfrac{1}{4}}{s-4}$

Finally as a function of t ;

$L^{-1} \begin {pmatrix} \dfrac{s}{((s-20)(s-5)(s-4)} \end {pmatrix}= \dfrac{e^{4t}}{12}\begin {pmatrix}e^{16t} - 4e^t+3 \end {pmatrix}$