1. \( \log\left( \frac{s + 1}{s} \right) \)
2. \( \log\left( \frac{s(s + 1)}{s^2 + 4} \right) \)
3. \( \cot^{-1}(s + 1) \)
4. \( \tan^{-1} \left( \frac{2}{s} \right) \)
5. \( \log \left( \frac{s + b}{s + a} \right) \)
6. \( \log \left( 1 - \frac{a^2}{s^2} \right) \)
1. \( \frac{1}{s^2(s^2 + 2)} \)
2. \( \frac{s}{(s^2 + a^2)^2} \)
3. \( \frac{1}{s(s^2 + 2)} \)
4. \( \frac{1}{s(s + 3)^2} \)
5. \( \frac{s^2}{(s^2 + 4)^2} \)
6. \( \frac{s}{s(s + 1)} \)
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. \( \frac{2s^2 - 1}{(s + 1)^2 (s^2 + 1)} \)
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. \( \frac{e^{-\pi s}}{s + 3} \)
If you want notes for a few difficult problems, click the link in the comments.
1. \( y'' - 5y' + 6y = 0,\quad y(0) = 0,\quad y'(0) = 1 \)
2. \( y'' + 4y' + 3y = e^{-t},\quad y(0) = 1,\quad y'(0) = 1 \)
3. \( \frac{d^4 y}{dt^4} - 81y = 0,\quad y(0) = 1,\quad y'(0) = y''(0) = y^{(3)}(0) = 0 \)
4. \( \frac{d^2 y}{dx^2} + \frac{dy}{dx} - 2y = x,\quad y(0) = 1,\quad y'(0) = 0 \)
5. \( \frac{d^2 y}{dt^2} + 2 \frac{dy}{dt} + y = \sin(t),\quad y(0) = 0,\quad y'(0) = 0 \)
6. \( y'' - 3y' + 2y = 1 - e^{2t},\quad y(0) = 1,\quad y'(0) = 0 \)
7. \( \frac{d^2 y}{dt^2} + 2 \frac{dy}{dt} + 5x = e^t \sin(t),\quad x(0) = 0,\quad x'(0) = 1 \)