Question 2. Find the inverse Laplace transform of the given function. a) \( \mathcal{L}^{-1}\left\{\frac{1}{s^{2}}-\frac{3}{s}+\frac{1}{s+2}\right\} \), b) \( \mathcal{L}^{-1}\left\{\frac{1}{4 s^{2}+1}\right\} \) c) \( \mathcal{L}^{-1}\left\{\left(\frac{2}{s}+\frac{1}{s^{3}}\right)^{2}\right\} \) d) \( \mathcal{L}^{-1}\left\{\frac{1}{s(s-1)\left(s^{2}+6 s+13\right)}\right\} \) e) \( \mathcal{L}^{-1}\left\{\frac{s}{(s+3)\left(s^{2}+4\right)}\right\} \), f) \( \mathcal{L}^{-1}\left\{\frac{s+1}{s^{2}-4 s}\right\} \)

Step1/4a) we have to find inverse laplce transform of \({1/s^2-3/s+1/(s+2)}\)\(L^-1{1/s^2-3/s+1/(s+2)}=L^{-1}\{\frac{1}{s^2}\}-L^{-1}\{\frac{3}{s}\}+L^{-1}\{\frac{1}{s+2}\}\)\(=t-3\text{H}\(t\)+e^{-2t}\)Step2/4b) we have to find laplace transform of \({1/(4s^2+1)}\)\(L^-1{1/(4s^2+1)}=\frac{1}{2}L^{-1}\{\frac{\frac{1}{2}}{s^2+\(\frac{1}{2}\)^2}\}\)\(=\frac{1}{2}\sin \(\frac{t}{2}\)\)Step3/4c) we have to find inverse laplace transform of \({(2/s+1/s^3)^2}\)\(L^-1{(2/s+1/s^3)^2}=L^{-1}\{\frac{4}{s^2}+\frac{4}{s^4}+\frac{1}{s^6}\}\)\(=L^{-1}\{\frac{4}{s^2}\}+L^{-1}\{\frac{4}{s^4}\}+L^{-1}\{\frac{1}{s^6}\}\)\(=4t+\frac{2t^3}{3}+\frac{t^5}{120}\)Step4/4d) we have to find laplace transform of \(1/(s(s-1)(s^2+6s+13))\)\(L^-1{1/(s(s-1)(s^2+6s+13))}= L^{-1}\{-\frac{1}{13s}+\frac{1}{20\(s-1\)}+\frac{7s+29}{260\(s^2+6s+13\)}\}\)\(=-L^{-1}\{\frac{1}{13s}\}+L^{-1}\{\frac{1}{20\(s-1\)}\}+\frac{7}{260}L^{-1}\{\frac{s+3}{\(s+3\)^2+4}\}+\frac{2}{65}L^{-1}\{\frac{1}{\(s+3\)^2+4}\}\)\(=-\frac{1}{13}\text{H}\(t\)+\frac{1}{20}e^t+\frac{7}{260}e^{-3t}\cos \(2t\)+\frac{2}{65}e^{-3t}\frac{1}{2}\sin \(2t\)\)e) \(L^-1{s/((s+3)(s^2+4))}=L^{-1}\{\frac{3s}{13\(s^2+4\)}+\frac{4}{13\(s^2+4\)}-\frac{3}{13\(s+3\)}\}\)\(=-\frac{3}{13}L^{-1}\{\frac{1}{s+3}\}+\frac{3}{13}L^{-1}\{\frac{s}{s^2+4}\}+\frac{4}{13}L^{-1}\{\frac{1}{s^2+4}\}\)\(=-\frac{3}{13}e^{-3t}+\frac{3}{13}\cos \(2t\)+\frac{4}{13}\cdot \frac{1}{2}\sin \(2t\)\)\(f)L^-1{(s+1)/(s^2-4s)}=L^{-1}\{-\frac{1}{4s}+\frac{5}{4\(s-4\)}\}\)\(=-\frac{1}{4}\text{H}\(t\)+\frac{5}{4}e^{4t}\)Explanation:The answer of question is shown in above steps ...