 # Obtain the Laplace Transform of Lleft{t^3-t^2+4tright} Lleft{3e^{4t}-e^{-2t}right} e1s2kat26 2021-02-02 Answered
Obtain the Laplace Transform of
$L\left\{{t}^{3}-{t}^{2}+4t\right\}$
$L\left\{3{e}^{4t}-{e}^{-2t}\right\}$
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Step 1
Question (1) −
Let the given function be y −
$y=L\left\{{t}^{3}-{t}^{2}+4t\right\}$
Using laplace transformation formula −
$L\left[{t}^{n}\right]=\frac{\left(n!\right)}{{s}^{n+1}}$
then,
$y=F\left(s\right)=L\left\{t3-t2+4t\right\}$
$F\left(s\right)=\frac{3!}{{s}^{3+1}}-\frac{2!}{{s}^{2+1}}+4\frac{1!}{{s}^{1+1}}$
$F\left(s\right)=\frac{3\cdot 2\cdot 1}{{s}^{4}}-\frac{2\cdot 1}{{s}^{3}}+4\frac{1}{{s}^{2}}$
$F\left(s\right)=\frac{6}{{s}^{4}}-\frac{2}{{s}^{3}}+\frac{4}{{s}^{2}}$
Step 2
Question (2) −
Given function −
$F\left(s\right)=L\left\{3{e}^{4t}-{e}^{-2t}\right\}$
Using Laplace transformation formula −
$L\left[eat\right]=\frac{1}{s-a}$
$F\left(s\right)=3\left[\frac{1}{\left(s-4\right)}\right]-\left[\frac{1}{s-\left(-2\right)}\right]$
$F\left(s\right)=\frac{3}{\left(s-4\right)}-\frac{1}{\left(s+2\right)}$