1. \( \frac{e^{-\pi s}}{s^2 + 4} \)
2. \( \frac{e^{-7s}}{(s^4)^3} \)
3. \( \frac{s e^{-4s} + \pi e^{-2s}}{s^2 + \pi^2} \)
4. \( \frac{e^{-2s}}{(s + 1)(s^2 + 2s + 2)} \)
5. \( \frac{e^{-\pi s}}{s + 3} \)
Will fix all the doubts in class.
1. \( y'' - 5y' + 6y = 0, \quad y(0) = 0, \quad y'(0) = 1 \)
2. \( y'' + 4y' + 3y = e^{-t}, \quad y(0) = y'(0) = 1 \)
3. \( \frac{d^2 y}{dt^2} - 8y = 0, \quad y(0) = 1, \quad y'(0) = y''(0) = y^{(3)}(0) = 0 \)
4. \( \frac{d^2 y}{dx^2} + \frac{dy}{dx} - 3y = x, \quad y(0) = 1, \quad y'(0) = 0 \)
5. \( \frac{d^2 y}{dt^2} + 2 \frac{dy}{dt} + y = \sin t, \quad y(0) = y'(0) = 0 \)
6. \( y'' - 3y' + 2y = 1 - e^{-2t}, \quad y(0) = 1, \quad y'(0) = 0 \)
\( x^4 + 2x^3 + 5x = e^{-t} \sin t, \quad x(0) = x'(0) = x''(0) = x^{(3)}(0) = 0 \)
1. \( \frac{s^2 + 5 - 2}{s(s + 3)(s - 2)} \)
2. \( \frac{s^2 - 105 + 13}{(s - 7)(s^2 - 5s + 6)} \)
3. \( \frac{1}{s^2(s + 1)} \)
4. \( \frac{2s + 3}{s^2 + 5s - 6} \)
5. \( \frac{s + 1}{s(s + 1)} \)
6. \( \frac{s^2}{(s - 1)^3} \)
7. \( \frac{s^2}{(s^2 + a^2)(s^2 + b^2)} \)
8. \( \frac{2s^2 - 1}{(s + 1)^2 (s^2 + 1)} \)
9. \( \log\left( \frac{s + 1}{s} \right) \)
10. \( \log\left( \frac{s(s + 1)}{s^2 + 4} \right) \)
11. \( \cot^{-1}(s + 1) \)
12. \( \tan^{-1} \left( \frac{2}{s} \right) \)
13. \( \log \left( \frac{s + b}{s + a} \right) \)
14. \( \log \left( 1 - \frac{a^2}{s^2} \right) \)
15. \( \frac{1}{s^2(s^2 + 2)} \)
16. \( \frac{s}{(s^2 + a^2)^2} \)
17. \( \frac{1}{s(s^2 + 2)} \)
18. \( \frac{1}{s(s + 3)^2} \)
19. \( \frac{s^2}{(s^2 + 4)^2} \)
20. \( \frac{s}{s(s + 1)} \)