What is the inverse laplace transform of $\frac{{s}^{3}-{a}^{2}s}{({s}^{2}+{a}^{2}{)}^{2}}$

Charlie Conner
2022-10-08
Answered

What is the inverse laplace transform of $\frac{{s}^{3}-{a}^{2}s}{({s}^{2}+{a}^{2}{)}^{2}}$

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vakleraarrc

Answered 2022-10-09
Author has **6** answers

Partial fraction expansion

$$\frac{s}{{a}^{2}+{s}^{2}}-\frac{2{a}^{2}s}{{({a}^{2}+{s}^{2})}^{2}}$$

Notice that

$$\frac{d}{ds}\frac{{a}^{2}}{{a}^{2}+{s}^{2}}=-\frac{2{a}^{2}s}{{({a}^{2}+{s}^{2})}^{2}}$$

Use the transform table and the derivative rule to find the result as

$$\mathrm{cos}(at)-at\mathrm{sin}(at)$$

This is a somewhat difficult problem I think

$$\frac{s}{{a}^{2}+{s}^{2}}-\frac{2{a}^{2}s}{{({a}^{2}+{s}^{2})}^{2}}$$

Notice that

$$\frac{d}{ds}\frac{{a}^{2}}{{a}^{2}+{s}^{2}}=-\frac{2{a}^{2}s}{{({a}^{2}+{s}^{2})}^{2}}$$

Use the transform table and the derivative rule to find the result as

$$\mathrm{cos}(at)-at\mathrm{sin}(at)$$

This is a somewhat difficult problem I think

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