演習問題

1.

次の初期値問題の解のラプラス変換を求めよ．

(a)

$${ y^{\prime\prime} + 4y^{\prime} - 5y = e^{t}, \ y(0) = 0, \ y^{\prime}(0) = 1}$$

(b)

$${ y^{\prime\prime} + 4y^{\prime} + 4y = e^{-t}, \ y(0) = 1, \ y^{\prime}(0) = 1}$$

(c)

$${ y^{\prime\prime} + 4y^{\prime} - 5y = f(t), \ y(0) = 0, \ y^{\prime}(0) = 0}$$

$$f(t) = \left\{\begin{array}{rl} 2,&0 < t < \pi\\ \cos{t},&t > \pi \end{array} \right.$$