91Ó°ÊÓ

Perform the appropriate partial fraction decomposition, and then use the result to find the inverse Laplace transform of the given function. $$ Y(s)=\frac{2 s^{2}+9 s+11}{(s+1)\left(s^{2}+4 s+5\right)} $$

Perform the appropriate partial fraction decomposition, and then use the result to find the inverse Laplace transform of the given function. $$ Y(s)=\frac{7 s^{2}+20 s+53}{(s-1)\left(s^{2}+2 s+5\right)} $$

Perform the appropriate partial fraction decomposition, and then use the result to find the inverse Laplace transform of the given function. $$ Y(s)=\frac{1}{(s-2)^{2}(s+1)^{3}} $$

Perform the appropriate partial fraction decomposition, and then use the result to find the inverse Laplace transform of the given function. $$ Y(s)=\frac{1}{(s-1)^{2}\left(s^{2}+4\right)} $$

In mathematics, the symbolism \([t]\) calls for the greatest integer not exceeding the number \(t\). For example, \([3.1]=3\), \([3]=3\), and \([-1.2]=-2\). Define $$ f(t)= \begin{cases}{[t],} & t \geq 0 \\ 0, & \text { otherwise }\end{cases} $$

Perform the appropriate partial fraction decomposition, and then use the result to find the inverse Laplace transform of the given function. $$ Y(s)=\frac{1}{(s+2)^{2}\left(s^{2}+9\right)} $$

Perform the appropriate partial fraction decomposition, and then use the result to find the inverse Laplace transform of the given function. $$ Y(s)=\frac{1}{(s-1)^{2}\left(s^{2}-9\right)} $$

In Exercises 34-41, use the propositions in Section 2 to transform the given initial value problem into an algebraic equation involving \(\mathcal{L}(y)\). Solve the resulting equation for the Laplace transform of \(y\). \(y^{\prime}+2 y=t \sin t, y(0)=1\)

Perform the appropriate partial fraction decomposition, and then use the result to find the inverse Laplace transform of the given function. $$ Y(s)=\frac{1}{(s+1)^{2}\left(s^{2}-4\right)} $$

Perform the appropriate partial fraction decomposition, and then use the result to find the inverse Laplace transform of the given function. $$ Y(s)=\frac{s}{(s+2)^{2}\left(s^{2}+9\right)} $$