演習問題

1. 次の微分方程式を解け．
$$\begin{array}{ll} (a) y^{\prime}\cos{x} - y\sin{x} + e^{x} ... ...{x} & (d) xy^{\prime} + (1 + x)y = e^{-x}\sin{2x} \end{array}$$
2. 次の初期値問題を解け．
$(a) y^{\prime} + (\cos{x})y = e^{-\sin{x}}, y(0) = 2$
$(b) (x\log{x})y^{\prime} - y = \log{x}, y(e) = -1$
$(c) y^{\prime} + y = f(x), y(0) = 0, f(x) = \left\{\begin{array}{ll} 1, & 0 \leq x < 1\\ 0, & x \geq 1 \end{array} \right.$
$(d) y^{\prime} + (\tan{x})y = \cos^{2}{x}, y(0) = -1$
3. $R = 10\Omega, E = 20V, L = \left\{\begin{array}{rl} 5-t,& 0 \leq t\leq 5\\ 0,& 5 \leq t \end{array} \right.$の$RL$回路で$i(0) = 0$のとき,$i(t)$を求めよ．