演習問題

1.

次の微分方程式を解け．

(a)

$${ y^{\prime}\cos{x} - y\sin{x} + e^{x} = 0}$$

(b)

$${ y^{\prime} + 2xy = 2x}$$

(c)

$${ xy^{\prime} + y = x\sin{x}}$$

(d)

$${ xy^{\prime} + (1 + x)y = e^{-x}\sin{2x}}$$

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．