(Solved): Solve the following differential equations using Laplace Transforms, using the posted table 3.1, a ...

Solve the following differential equations using Laplace Transforms, using the posted table $$3.1$$, and with all derivatives of $$f^{??}(0)=f^{?}(0)=f(0)=0$$. (d) $$6\left(d^{2}f/d{t}^2\right)+5(df/dt)+f(t)=9$$ (e) $$16\left(d^{2}f/d{t}^2\right)+4(df/dt)+f(t)=10$$ Table 3.1 Laplace Transforms for Various Time-Domain Functions* 14. $$\sin ?t$$ 15. $$\cos ?t$$ 16. $$\sin (?t+?)$$ $$\frac{?}{s^2+{?}^2}$$ $$\begin{array}{l}\text{ 17. }{e}^{?bt}\sin ?t\\ \text{ 18. }{e}^{?bt}\sin ?t\end{array}\}b,?$$ real $$\frac{s}{s^2+{?}^2}$$ $$\frac{?\cos ?+s\sin ?}{s^2+{?}^2}$$ 19. $$\frac{1}{?\sqrt{1?{?}^2}}{e}^{??t/?}\sin \left(\sqrt{1?{?}^2}t/?\right)$$ $$\begin{array}{l}\frac{?}{(s+b{)}^2+{?}^2}\\ \frac{s+b}{(s+b{)}^2+{?}^2}\end{array}$$ $$(0????<1)$$ $$20.1+\frac{1}{{?}_2?{?}_1}\left({?}_1{e}^{?t/{?}_1}?{?}_2{e}^{?t/{?}_2}\right)$$ $$\frac{1}{{?}^2{s}^2+2??s+1}$$ $$\left({?}_1?={?}_2\right)$$ 21. $$1?\frac{1}{\sqrt{1?{?}^2}}{e}^{??t/?}\sin \left[\sqrt{1?{?}^2}t/?+?\right]$$ $$\frac{1}{s\left({?}^2{s}^2+2??s+1\right)}$$ $$?={\tan }^{?1}\frac{\sqrt{1?{?}^2}}{?},(0????<1)$$ Tohla 21 (Continuod) *Note that $$f(t)$$ and $$F(s)$$ are defined for $$t?0$$ only.