Question Number 194887 by Mastermind last updated on 18/Jul/23
$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{Inverse}\:\mathrm{laplace}\:\mathrm{transform}\:\mathrm{of} \\ $$$$\frac{\mathrm{S}\:+\:\mathrm{2}}{\mathrm{S}^{\mathrm{2}} \:+\mathrm{4S}\:+\:\mathrm{7}} \\ $$$$ \\ $$$$\mathrm{Urgent}! \\ $$
Answered by Dwan last updated on 19/Jul/23
$$\frac{{s}+\mathrm{2}}{{s}^{\mathrm{2}} +\mathrm{4}{s}+\mathrm{7}} \\ $$$$=\frac{{s}+\mathrm{2}}{\left({s}+\mathrm{2}\right)^{\mathrm{2}} +\mathrm{3}} \\ $$$$\mathscr{L}\left({cosat}\right)={Re}\left(\int_{\mathrm{0}} ^{\infty} {e}^{−{st}} {e}^{{iat}} {dt}\right)={Re}\left(\frac{\mathrm{1}}{{s}−{ia}}\right) \\ $$$$=\frac{{s}}{{s}^{\mathrm{2}} +{a}^{\mathrm{2}} } \\ $$$$\mathscr{L}^{−\mathrm{1}} \left(\frac{{s}+\mathrm{2}}{\left({s}+\mathrm{2}\right)^{\mathrm{2}} +\mathrm{3}}\right)={cos}\left(\sqrt{\mathrm{3}}{t}\right) \\ $$