Attempting the following partial fraction equation and was wondering how to approach the $s^2$ outside the brackets: $$\frac{1}{s^2(s^2+2s+10)}$$
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In what problem did you encounter this? Maybe we have a better method of solving it that doesn't involve this. See also How to ask a Good Question. – DynamoBlaze Dec 04 '17 at 11:23
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The question asks to find the inverse laplace transform of the above equation. – Matlab rookie Dec 04 '17 at 11:27
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They're not "terms", but "factors." And they're different. One is an irreducible quadratic. The other is a linear squared. If it were written $(s-0)^2$, you might not have been confused(?) – B. Goddard Dec 04 '17 at 11:40
2 Answers
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$$\frac{1}{s^2(s^2+2s+10)}=\frac{A}{s}+\frac{B}{s^2}+\frac{Cs+D}{s^2+2s+10}$$
Now, just cross-multiply and get the values of $A,B,C,D$ in the usual way.
Your IDE
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would you use a similiar method if the 1 was substituted for $e^{-s}$? – Matlab rookie Dec 04 '17 at 11:41
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Sure. For $e^{-s}$ Laplace inverse would be just $f(t-1)$ in time domain shift. – Your IDE Dec 04 '17 at 11:48
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Follow the help given by @Your IDE and the answer will be:
$$\frac{s-3}{50 \left(s^2+2 s+10\right)}+\frac{1}{10 s^2}-\frac{1}{50 s}$$
Enrico M.
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