# Find the laplace transform of the following: Change of Scale text{If } Lleft{f(t)right}=frac{s^2-s+1}{(2s+1)^2(s-2)} text{ , find } Lleft{f(2t)right}

Find the laplace transform of the following:
Change of Scale
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brawnyN
$\text{Step 1}$
$\text{According to change of scale property in Laplace transform}$

$L\left\{f\left(at\right)\right\}=\frac{1}{a}F\left(\frac{s}{a}\right)$
$\text{Given that}$
$L\left\{f\left(t\right)\right\}=\frac{{s}^{2}-s+1}{\left(2s+1{\right)}^{2}\left(s-2\right)}$
$\text{Step 2}$

$L\left\{f\left(2t\right)\right\}=\frac{1}{2}\frac{\left(\frac{s}{2}{\right)}^{2}-\frac{s}{2}+1}{\left(2\frac{s}{2}+1{\right)}^{2}\left(\frac{s}{2}-1\right)}$
$=\frac{1}{2}×\frac{\frac{{s}^{2}}{4}-\frac{s}{2}+1}{\left(s+1{\right)}^{2}\left(\frac{s-2}{2}\right)}$
$=\frac{1}{2}×\frac{\frac{{s}^{2}-2s+4}{4}}{\frac{\left(s+1{\right)}^{2}\left(s-2\right)}{2}}$
$=\frac{1}{4}\cdot \frac{{s}^{2}-2s+4}{\left(s+1{\right)}^{2}\left(s-2\right)}$