Solution:- \(L\)\(({t^{2}e^{2t}+2\sin(t)\cos(t)+3})\) \(=\)\(L(t^{2}e^{2t})\) +\(L\)\((2\sin(t)\cos(t))\) +\(L(3)\) ——————————-(\(1)\) \(L(t^{2}e^{2t})\)\(=L(t^{2})\).\(L(e^{2t})\) \(=\frac{2}{s^3}.\frac{1}{S-2}\), ——————————————————–(\(a\)) Since, \(L({t^n})\) \(=\) \(\frac{n!}{S^{n+1}}\) \(\Rightarrow\)\(L(t^{2})\)\(=\) \(\frac{2}{S^{2+1}}\) \(=\) \(\frac{2}{S^{3}}\) Since, \(L(e^{at})\)\(=\)\(\frac{1}{(S-a)}\)\(=\)\(\frac{1}{S-2}\) Now, \(L({2\sin(t)\cos(t)})\) Which can be written as \(=L({\sin2(t)})\) \(L(\sin2t)\)\(=\)\(\frac{2}{S^{2}+2^{2}}\) \(=\)\(\frac{2}{(S^{2}+4)}\) ———————————————————–(\(b\)) Since, Laplace transformation of \(L\)\((\sin\)\(at\)) \(=\)\(\frac{a}{S^{2}+a^{2}}\) now, \(L(3)\)\(=\frac{3}{S}\) ———————————————————-(\(c\)) Since, \(L(1)\)\(=\frac{1}{S}\) Now putting the values of (\(a)\), Now putting the values of (\(a)\), Now putting […]