演習問題

1.

次の微分方程式を解け．

(a)

$${ \left\{\begin{array}{rl} x_{1}^{\prime} =& 2x_{1} + x_{2} - e^{t}\\ x_{2}^{\prime} =& 3x_{1} + 4x_{2} - 7e^{t} \end{array} \right . }$$

(b)

$${ {\bf X}^{\prime} = \left(\begin{array}{cc} 0&4\\ -1&0 \end{array}\right){\bf X} + \left(\begin{array}{c} -3\cos{t}\\ 0 \end{array}\right) }$$

(c)

$${ \left\{\begin{array}{rl} x_{1}^{\prime} =& x_{1} + x_{2} + x_{3}... ...2}\\ x_{3}^{\prime} =& -2x_{1} -x_{2} -2x_{3} + 3e^{-t} \end{array} \right . }$$

(d)

$${ {\bf X}^{\prime} = \left(\begin{array}{rrrr} 0&1&0&0\\ 0&0&1&0\... ...right){\bf X} + \left(\begin{array}{c} t^{2}\\ 0\\ 0\\ 0 \end{array}\right)}$$

2.

次の微分方程式を連立微分方程式に直して解け．

$$y^{\prime\prime} + 4y^{\prime} + 3y = t$$

3.

次の微分方程式を消去法を用いて解け．

(a)

$${ \left\{\begin{array}{rl} x_{1}^{\prime} + x_{2}^{\prime} - 2x_{1... ...^{t}\\ x_{1}^{\prime} + x_{2}^{\prime} - x_{2} &= e^{4t} \end{array}\right . }$$

(b)

$${ \left\{\begin{array}{rl} x_{1}^{\prime\prime} + x_{2}^{\prime} -... ..._{1}^{\prime} + x_{2}^{\prime\prime} - x_{1} + x_{2} &= 0 \end{array}\right . }$$

(c)

$${ \left\{\begin{array}{rl} 2x_{1}^{\prime} + x_{2}^{\prime} + x_{1... ...\ x_{1}^{\prime} + x_{2}^{\prime} + 2x_{1} + 2x_{2} &= 2 \end{array}\right . }$$

(d)

$${ \left\{\begin{array}{rl} x_{1}^{\prime\prime} - x_{2}^{\prime} &... ...x_{1}^{\prime} + x_{2}^{\prime} - 3x_{1} + x_{2} &= 2t - 1 \end{array}\right .}$$