Find the inverse Laplace transform of $$\frac{1}{{({s}^{2}+1)}^{2}}$$

propappeale00
2022-10-12
Answered

Find the inverse Laplace transform of $$\frac{1}{{({s}^{2}+1)}^{2}}$$

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occuffick24

Answered 2022-10-13
Author has **13** answers

Hint:

$${\mathcal{L}}^{-1}\left[{\displaystyle \frac{1}{({s}^{2}+{\omega}^{2}{)}^{2}}}\right]={\displaystyle \frac{1}{2{\omega}^{3}}}(\mathrm{sin}\omega t-\omega t\mathrm{cos}\omega t)$$

Note: $\omega $ is a real constant in this generalization.

Now, can you find the Laplace transform of $\mathrm{sin}\omega t$ and $\omega t\mathrm{cos}\omega t$ to understand what is going on with one of the shift theorems and why this is the result?

Of course you can always use the formal definitions to find this also if that is the approach required.

$${\mathcal{L}}^{-1}\left[{\displaystyle \frac{1}{({s}^{2}+{\omega}^{2}{)}^{2}}}\right]={\displaystyle \frac{1}{2{\omega}^{3}}}(\mathrm{sin}\omega t-\omega t\mathrm{cos}\omega t)$$

Note: $\omega $ is a real constant in this generalization.

Now, can you find the Laplace transform of $\mathrm{sin}\omega t$ and $\omega t\mathrm{cos}\omega t$ to understand what is going on with one of the shift theorems and why this is the result?

Of course you can always use the formal definitions to find this also if that is the approach required.

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