Question Number 180145 by ali009 last updated on 07/Nov/22
![determine 1)L^− [((4s^2 −17s−24)/(s(s+3)(s−4)))] 2)L^− [((5s^2 −4s−7)/((s−3)(s^2 +4)))]](https://www.tinkutara.com/question/Q180145.png)
$${determine} \\ $$$$\left.\mathrm{1}\right)\mathcal{L}^{−} \left[\frac{\mathrm{4}{s}^{\mathrm{2}} −\mathrm{17}{s}−\mathrm{24}}{{s}\left({s}+\mathrm{3}\right)\left({s}−\mathrm{4}\right)}\right] \\ $$$$\left.\mathrm{2}\right)\mathcal{L}^{−} \left[\frac{\mathrm{5}{s}^{\mathrm{2}} −\mathrm{4}{s}−\mathrm{7}}{\left({s}−\mathrm{3}\right)\left({s}^{\mathrm{2}} +\mathrm{4}\right)}\right] \\ $$
Answered by Ar Brandon last updated on 08/Nov/22
$$\mathcal{L}^{−\mathrm{1}} \left(\frac{\mathrm{4}{s}^{\mathrm{2}} −\mathrm{17}{s}−\mathrm{24}}{{s}\left({s}+\mathrm{3}\right)\left({s}−\mathrm{4}\right)}\right) \\ $$$$=\mathcal{L}^{−\mathrm{1}} \left(\frac{\mathrm{2}}{{s}}+\frac{\mathrm{3}}{{s}+\mathrm{3}}−\frac{\mathrm{1}}{{s}−\mathrm{4}}\right) \\ $$$$=\mathrm{2}+\mathrm{3}{e}^{−\mathrm{3}{t}} −{e}^{\mathrm{4}{t}} \\ $$$$ \\ $$$$\mathcal{L}^{−\mathrm{1}} \left(\frac{\mathrm{5}{s}^{\mathrm{2}} −\mathrm{4}{s}−\mathrm{7}}{\left({s}−\mathrm{3}\right)\left({s}^{\mathrm{2}} +\mathrm{4}\right)}\right) \\ $$$$=\mathcal{L}^{−\mathrm{1}} \left(\frac{\mathrm{2}}{{s}−\mathrm{3}}+\frac{\mathrm{3}{s}+\mathrm{5}}{{s}^{\mathrm{2}} +\mathrm{4}}\right) \\ $$$$=\mathcal{L}^{−\mathrm{1}} \left(\frac{\mathrm{2}}{{s}−\mathrm{3}}+\frac{\mathrm{3}{s}}{{s}^{\mathrm{2}} +\mathrm{4}}+\frac{\mathrm{5}}{{s}^{\mathrm{2}} +\mathrm{4}}\right) \\ $$$$=\mathrm{2}{e}^{\mathrm{3}{t}} +\mathrm{3cos}\left(\mathrm{2}{t}\right)+\frac{\mathrm{5}}{\mathrm{2}}\mathrm{sin}\left(\mathrm{2}{t}\right) \\ $$
Commented by ali009 last updated on 08/Nov/22
$${thank}\:{you} \\ $$