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I just don't seem to be able to crack this inverse Laplace:

$$ \mathcal{L}^{-1} \left\{ \frac{20000s}{s^2 + 20000s + 5\cdot 10^8} \right\} (t). $$

Could someone help me out? I'm totally lost. I don't seem to be able to partial fraction decompose it, so I don't know where to start.

Gary
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guywithaquestion
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2 Answers2

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Hint

$(s+10000-20000i)(s+10000+20000i)=s^2+20000s+5\cdot 10^8$

amWhy
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Meow
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Noting $$ s^2+20000s+5\cdot 10^8=(s+10000)^2+20000^2 $$ one has $$ L^{-1}\bigg(\frac{20000s}{s^2+20000s+5\cdot 10^8}\bigg)=L^{-1}\bigg(\frac{20000(s+10000)-2\cdot10000^2}{(s+10000)^2+20000^2}\bigg)=20000e^{-10000t}\cos(20000t)-10000e^{-10000t}\sin(20000t). $$ Here $$ L^{-1}\bigg(\frac{s-a}{(s-a)^2+b^2}\bigg)=e^{at}\cos(bt), L^{-1}\bigg(\frac{b}{(s-a)^2+b^2}\bigg)=e^{at}\sin(bt).$$

xpaul
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  • I post the following to ensure you cannot claim ignorance in the future: Please read thoroughly, Enforcement of Quality Standards. Having done this, users can now flag ongoing occurrences of further answers from you to very low quality questions, if the pattern continues, for moderator attention, along with a link to this comment. Please do not answer low quality questions. – amWhy Nov 19 '21 at 21:28