Ejercicios de repaso sección 4.5

Encuentre $f(t)=\mathscr{L}^{-1}F(s)$ o $F(s)= \mathscr{L}f(t)$

  1. $\hspace{0.5cm} \mathscr{L}\left\{\left(t-4\right)\mu\left(t-4\right)\right.$

  2. $\hspace{0.5cm} \mathscr{L}\left\{e^{2-t}\mu\left(t-2\right)\right.$

  3. $\hspace{0.5cm} \mathscr{L}\left\{cos2t\mu\left(t-\pi\right)\right.$

  4. $\hspace{0.5cm} \mathscr{L}\left\{sent\mu\left(t-\dfrac{\pi}{2}\right)\right.$

  5. $\hspace{0.5cm} \mathscr{L}^{-1}\left\{\dfrac{e^{-2s}}{s^3}\right\}$

  6. $\hspace{0.5cm} \mathscr{L}^{-1}\left\{\dfrac{e^{-\pi s}}{s^2+1}\right\}$

  7. $\hspace{0.5cm} \mathscr{L}^{-1}\left\{\dfrac{se^{-\frac{\pi}{2}s}}{s^2+4}\right\}$

  8. $\hspace{0.5cm} \mathscr{L}^{-1}\left\{\dfrac{e^{-s}}{s\left(s+1\right)}\right\}$

  9. $\hspace{0.5cm} \mathscr{L}^{-1}\left\{\dfrac{e^{-2s}}{s^2\left(s-1\right)}\right\}$

    10. En los ejercicios 10 a 13, evaluar la función

  10. $\hspace{0.5cm} f(t)=\left\{\begin{matrix}t,&0\le t<1\\0,&t\geq1\\\end{matrix}\right.$

  11. $\hspace{0.5cm} f(t)=\left\{\begin{matrix}1,&0\le t<4\\0,&4\le t<5\\1,&t\geq5\\\end{matrix}\right.$

  12. $\hspace{0.5cm} f(t)=\left\{\begin{matrix}0,&0\le t<1\\t^2,&t\geq1\\\end{matrix}\right.$

  13. $\hspace{0.5cm} f(t)=\left\{\begin{matrix}sent,&0\le t<2\pi\\0,&t\geq2\pi\\\end{matrix}\right.$

    14. En los ejercicios 14 a 17 resolver el PVI

  14. $\hspace{0.5cm} y\prime+y=f(t)$ sujeta a la condición $y(0)=5$ donde $f(t)=\left\{\begin{matrix}0,&0\le t<\pi\\3cos{t},&t\geq\pi\\\end{matrix}\right.$

  15. $\hspace{0.5cm} y\prime+y=f(t)$ sujeta a la condición $y(0)=0$ donde $f(t)=\left\{\begin{matrix}0,&0\le t<1\\5,&t\geq1\\\end{matrix}\right.$

  16. $\hspace{0.5cm} y\prime\prime+4y=f(t)$ sujeta a la condición $y(0)=0 \;\; \wedge \;\; y´(0)=-1$ donde $f(t)=\left\{\begin{matrix}1,&0\le t<1\\0,&t\geq1\\\end{matrix}\right.$

  17. $\hspace{0.5cm} y\prime\prime+4y=sent\mu(t-2\pi)$ sujeta a la condición $y(0)=1 \;\; \wedge \;\; y´(0)=0$