2014年度秋学期　応用数学（解析）　第２部・基本的な微分方程式　／　第８回　２階線形微分方程式（２） (2014. 11. 13)

A. A

sano

, Kan

sai U

niv.

2014

A. A

sano

, Kan

sai U

niv.

A. A

sano

, Kan

sai U

niv.

2014

A. A

sano

, Kan

sai U

niv.

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

x(t) = et2et + aet + bet = 0(

2 + a+ b)et = 0

(3)

2 + a + b = 0 x(t) = et 2 + a+ b = 0 1,2 x(t) = C1e1t + C2e2tC1, C2 C1 = C2 = 0 x(t) 0

C1e1t + C2e2t x(t) = et

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014 (2014. 11. 6) http://racco.mikeneko.jp/ 1/5

2014

A. A

sano

, Kan

sai U

niv.

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014

A. A

sano

, Kan

sai U

niv.

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

0

2014

A. A

sano

, Kan

sai U

niv.

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

0

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014

A. A

sano

, Kan

sai U

niv.

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

0

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014

A. A

sano

, Kan

sai U

niv.

x 0

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

0

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014

A. A

sano

, Kan

sai U

niv.

x 0

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

0

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014

A. A

sano

, Kan

sai U

niv.

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014

A. A

sano

, Kan

sai U

niv.

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014

A. A

sano

, Kan

sai U

niv.

0 x = et

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014

A. A

sano

, Kan

sai U

niv.

0 x = et

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

1, 2

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014

A. A

sano

, Kan

sai U

niv.

0 x = et

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

1, 2

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

x = C1e1t + C2e2t

2014

A. A

sano

, Kan

sai U

niv.

0 x = et

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

1, 2

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

x = C1e1t + C2e2t x 0

2014

A. A

sano

, Kan

sai U

niv.

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

1(t), 2(t), . . . , n(t)c11(t)+ c22(t)+ + cnn(t) = 0t = t0c11(t0) + c22(t0) + + cnn(t0) =c1e1 + c2e2 + + cnen = 0 e1, e2, . . . , en c1 = c2 = = cn = 0 1(t), 2(t), . . . , n(t) x11(t)+ x22(t)+ + xnn(t)t = t0 x1e1 + x2e2 + + xnen x(t) x11(t) + x22(t) + + xnn(t) t = t0 x1e1 + x2e2 + + xnen 1.

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x + ax + bx = 02 + a+ b = 0 x(t) = et 2 + a+ b = 01. 1,2 e1t, e2t x(t) = C1e1t + C2e2tC1, C22. + i, i x(t) = C1e(+i)t + C2e(i)t

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

)= et (C1(cos(t) + i sin(t)) + C2(cos(t) i sin(t)))= et ((C1 + C2) cos(t) + i(C1 C2) sin(t))

(9)

x(t) = et (C1 cos(t) + C2 sin(t)) 13. 1C1e1t te1t

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x = et

2014

A. A

sano

, Kan

sai U

niv.

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x = et

2014

A. A

sano

, Kan

sai U

niv.

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x = et

2014

A. A

sano

, Kan

sai U

niv.

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x = et

2014

A. A

sano

, Kan

sai U

niv.

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

2014

A. A

sano

, Kan

sai U

niv.

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

2014

A. A

sano

, Kan

sai U

niv.

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

2014

A. A

sano

, Kan

sai U

niv.

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

2014

A. A

sano

, Kan

sai U

niv.

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

2014

A. A

sano

, Kan

sai U

niv.

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

2014

A. A

sano

, Kan

sai U

niv.

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

2014

A. A

sano

, Kan

sai U

niv.

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

2014

A. A

sano

, Kan

sai U

niv.

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

2014

A. A

sano

, Kan

sai U

niv.

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

2014

A. A

sano

, Kan

sai U

niv.

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

2014

A. A

sano

, Kan

sai U

niv.

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

2014

A. A

sano

, Kan

sai U

niv.

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

2014

A. A

sano

, Kan

sai U

niv.

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

2014

A. A

sano

, Kan

sai U

niv.

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

(21t+ 21)e

1t + a1te1t + bte1t

= {21 + a1 + b}te1t + (21 + a)e1t(11)

1 {} 0 21 = a() 0te1t e1t C1e1t + C2te1tC1, C2te1t C1e1tC1 t C1(t)

(C1e1t) = C 1e

1t + 1C1e1t = (C 1 + 1C1)e

1t

(C1e1t) = (C 1 + 1C

1)e

1t + 1(C1 + 1C1)e

1t = (C 1 + 21C1 +

21C1)e

1t(12)

(C 1 + 21C

1 +

21C1)e

1t + a(C 1 + 1C1)e1t + bC1e

1t = 0

(C 1 + 21C1 +

21C1) + a(C

1 + 1C1) + bC1 = 0

(13)

a2 4b = 0b = a2

4

21 = a 1 = a2

(C 1 aC 1 +a2

4C1) + a(C

1 +

a

2C1) +

a2

4C1 = 0 (14)

C 1 (t) = 0C1(t) C1(t) = pt+ qp, q (pt + q)e1tC1e1t + (pt+ q)e1tC1e1t +C2te1tC1, C2

x(t) x 5x + 6x = 0 x(0) = 1, x(0) = 0 2 5+ 6 = 0 = 2, 3 x(t) = C1e2t + C2e3tC1, C2x(0) = C1 +C2 = 1, x(0) = 2C1 + 3C2 = 0 C1 = 3, C2 = 2 3e2t 2e3t

x(t)

2014

A. A

sano

, Kan

sai U

niv.

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

(21t+ 21)e

1t + a1te1t + bte1t

= {21 + a1 + b}te1t + (21 + a)e1t(11)

(C1e1t) = C 1e

1t + 1C1e1t = (C 1 + 1C1)e

1t

(C1e1t) = (C 1 + 1C

1)e

1t + 1(C1 + 1C1)e

1t = (C 1 + 21C1 +

21C1)e

1t(12)

(C 1 + 21C

1 +

21C1)e

1t + a(C 1 + 1C1)e1t + bC1e

1t = 0

(C 1 + 21C1 +

21C1) + a(C

1 + 1C1) + bC1 = 0

(13)

a2 4b = 0b = a2

4

21 = a 1 = a2

(C 1 aC 1 +a2

4C1) + a(C

1 +

a

2C1) +

a2

4C1 = 0 (14)

x(t)

2014

A. A

sano

, Kan

sai U

niv.

1 0

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

(21t+ 21)e

1t + a1te1t + bte1t

= {21 + a1 + b}te1t + (21 + a)e1t(11)

(C1e1t) = C 1e

1t + 1C1e1t = (C 1 + 1C1)e

1t

(C1e1t) = (C 1 + 1C

1)e

1t + 1(C1 + 1C1)e

1t = (C 1 + 21C1 +

21C1)e

1t(12)

(C 1 + 21C

1 +

21C1)e

1t + a(C 1 + 1C1)e1t + bC1e

1t = 0

(C 1 + 21C1 +

21C1) + a(C

1 + 1C1) + bC1 = 0

(13)

a2 4b = 0b = a2

4

21 = a 1 = a2

(C 1 aC 1 +a2

4C1) + a(C

1 +

a

2C1) +

a2

4C1 = 0 (14)

x(t)

2014

A. A

sano

, Kan

sai U

niv.

1 0

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

(21t+ 21)e

1t + a1te1t + bte1t

= {21 + a1 + b}te1t + (21 + a)e1t(11)

(C1e1t) = C 1e

1t + 1C1e1t = (C 1 + 1C1)e

1t

(C1e1t) = (C 1 + 1C

1)e

1t + 1(C1 + 1C1)e

1t = (C 1 + 21C1 +

21C1)e

1t(12)

(C 1 + 21C

1 +

21C1)e

1t + a(C 1 + 1C1)e1t + bC1e

1t = 0

(C 1 + 21C1 +

21C1) + a(C

1 + 1C1) + bC1 = 0

(13)

a2 4b = 0b = a2

4

21 = a 1 = a2

(C 1 aC 1 +a2

4C1) + a(C

1 +

a

2C1) +

a2

4C1 = 0 (14)

x(t)

2014

A. A

sano

, Kan

sai U

niv.

1 0

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

(21t+ 21)e

1t + a1te1t + bte1t

= {21 + a1 + b}te1t + (21 + a)e1t(11)

(C1e1t) = C 1e

1t + 1C1e1t = (C 1 + 1C1)e

1t

(C1e1t) = (C 1 + 1C

1)e

1t + 1(C1 + 1C1)e

1t = (C 1 + 21C1 +

21C1)e

1t(12)

(C 1 + 21C

1 +

21C1)e

1t + a(C 1 + 1C1)e1t + bC1e

1t = 0

(C 1 + 21C1 +

21C1) + a(C

1 + 1C1) + bC1 = 0

(13)

a2 4b = 0b = a2

4

21 = a 1 = a2

(C 1 aC 1 +a2

4C1) + a(C

1 +

a

2C1) +

a2

4C1 = 0 (14)

x(t)

0

2014

A. A

sano

, Kan

sai U

niv.

1 0

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

(21t+ 21)e

1t + a1te1t + bte1t

= {21 + a1 + b}te1t + (21 + a)e1t(11)

(C1e1t) = C 1e

1t + 1C1e1t = (C 1 + 1C1)e

1t

(C1e1t) = (C 1 + 1C

1)e

1t + 1(C1 + 1C1)e

1t = (C 1 + 21C1 +

21C1)e

1t(12)

(C 1 + 21C

1 +

21C1)e

1t + a(C 1 + 1C1)e1t + bC1e

1t = 0

(C 1 + 21C1 +

21C1) + a(C

1 + 1C1) + bC1 = 0

(13)

a2 4b = 0b = a2

4

21 = a 1 = a2

(C 1 aC 1 +a2

4C1) + a(C

1 +

a

2C1) +

a2

4C1 = 0 (14)

x(t)

0

(21t+ 21)e

1t + a1te1t + bte1t

= {21 + a1 + b}te1t + (21 + a)e1t(11)

(C1e1t) = C 1e

1t + 1C1e1t = (C 1 + 1C1)e

1t

(C1e1t) = (C 1 + 1C

1)e

1t + 1(C1 + 1C1)e

1t = (C 1 + 21C1 +

21C1)e

1t(12)

(C 1 + 21C

1 +

21C1)e

1t + a(C 1 + 1C1)e1t + bC1e

1t = 0

(C 1 + 21C1 +

21C1) + a(C

1 + 1C1) + bC1 = 0

(13)

a2 4b = 0b = a2

4

21 = a 1 = a2

(C 1 aC 1 +a2

4C1) + a(C

1 +

a

2C1) +

a2

4C1 = 0 (14)

x(t)

2014

A. A

sano

, Kan

sai U

niv.

1 0

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

x(t) = x11(t) + x22(t) + + xnn(t) (8) n

x(t) = C1e(+i)t + C2e

(i)t

= et(C1e

it + C2eit

(9)

(te1t) = 1te1t + e1t = (1t+ 1)e1t

(te1t) = 1(1t+ 1)e1t + 1e1t = (21t+ 21)e1t

(10)

1

(21t+ 21)e

1t + a1te1t + bte1t

= {21 + a1 + b}te1t + (21 + a)e1t(11)

(C1e1t) = C 1e

1t + 1C1e1t = (C 1 + 1C1)e

1t

(C1e1t) = (C 1 + 1C

1)e

1t + 1(C1 + 1C1)e

1t = (C 1 + 21C1 +

21C1)e

1t(12)

(C 1 + 21C

1 +

21C1)e

1t + a(C 1 + 1C1)e1t + bC1e

1t = 0

(C 1 + 21C1 +

21C1) + a(C

1 + 1C1) + bC1 = 0

(13)

a2 4b = 0b = a2

4

21 = a 1 = a2

(C 1 aC 1 +a2

4C1) + a(C

1 +

a

2C1) +

a2

4C1 = 0 (14)

x(t)

0

(21t+ 21)e

1t + a1te1t + bte1t

= {21 + a1 + b}te1t + (21 + a)e1t(11)

(C1e1t) = C 1e

1t + 1C1e1t = (C 1 + 1C1)e

1t

(C1e1t) = (C 1 + 1C

1)e

1t + 1(C1 + 1C1)e

1t = (C 1 + 21C1 +

21C1)e

1t(12)

(C 1 + 21C

1 +

21C1)e

1t + a(C 1 + 1C1)e1t + bC1e

1t = 0

(C 1 + 21C1 +

21C1) + a(C

1 + 1C1) + bC1 = 0

(13)

a2 4b = 0b = a2

4

21 = a 1 = a2

(C 1 aC 1 +a2

4C1) + a(C

1 +

a

2C1) +

a2

4C1 = 0 (14)

x(t)

A. A

sano

, Kan

sai U

niv.

A. A

sano

, Kan

sai U

niv.

2014

A. A

sano

, Kan

sai U

niv.

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014

A. A

sano

, Kan

sai U

niv.

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014

A. A

sano

, Kan

sai U

niv.

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014

A. A

sano

, Kan

sai U

niv.

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014 (2)

x(t)x+ax+bx = 0a, bx+ax+bx = R(t)n x = A(t)x+ b(t)

x = A(t)x+ b(t) (1)

xp(t) b(t) 0

x = A(t)x (2)

xh(t)(1)

xs(t) = xh(t) + xp(t) (3)

(3) xs(t) (1)(1)

A(t)xs(t) + b(t)

= A(t) (xh(t) + xp(t)) + b(t)

= (A(t)xh(t)) + (A(t)xp(t) + b(t))

= (xh(t)) + (xp(t)) = (xs(t))

(4)

xs(t) (1) xs(t) (3)(1)xs(t0) = x0xh(t)(2)(2)

xh(t0) = x0 xp(t0) (5)

(2)

2014

A. A

sano

, Kan

sai U

niv.

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014 (2)

x = A(t)x+ b(t) (1)

xp(t) b(t) 0

x = A(t)x (2)

xh(t)(1)

xs(t) = xh(t) + xp(t) (3)

(3) xs(t) (1)(1)

A(t)xs(t) + b(t)

= A(t) (xh(t) + xp(t)) + b(t)

= (A(t)xh(t)) + (A(t)xp(t) + b(t))

= (xh(t)) + (xp(t)) = (xs(t))

(4)

xs(t) (1) xs(t) (3)(1)xs(t0) = x0xh(t)(2)(2)

xh(t0) = x0 xp(t0) (5)

(2)

2014

A. A

sano

, Kan

sai U

niv.

t

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014 (2)

x = A(t)x+ b(t) (1)

xp(t) b(t) 0

x = A(t)x (2)

xh(t)(1)

xs(t) = xh(t) + xp(t) (3)

(3) xs(t) (1)(1)

A(t)xs(t) + b(t)

= A(t) (xh(t) + xp(t)) + b(t)

= (A(t)xh(t)) + (A(t)xp(t) + b(t))

= (xh(t)) + (xp(t)) = (xs(t))

(4)

xs(t) (1) xs(t) (3)(1)xs(t0) = x0xh(t)(2)(2)

xh(t0) = x0 xp(t0) (5)

(2)

2014

A. A

sano

, Kan

sai U

niv.

t

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014 (2)

x = A(t)x+ b(t) (1)

xp(t) b(t) 0

x = A(t)x (2)

xh(t)(1)

xs(t) = xh(t) + xp(t) (3)

(3) xs(t) (1)(1)

A(t)xs(t) + b(t)

= A(t) (xh(t) + xp(t)) + b(t)

= (A(t)xh(t)) + (A(t)xp(t) + b(t))

= (xh(t)) + (xp(t)) = (xs(t))

(4)

xs(t) (1) xs(t) (3)(1)xs(t0) = x0xh(t)(2)(2)

xh(t0) = x0 xp(t0) (5)

(2)

2014

A. A

sano

, Kan

sai U

niv.

t

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014 (2)

x = A(t)x+ b(t) (1)

xp(t) b(t) 0

x = A(t)x (2)

xh(t)(1)

xs(t) = xh(t) + xp(t) (3)

(3) xs(t) (1)(1)

A(t)xs(t) + b(t)

= A(t) (xh(t) + xp(t)) + b(t)

= (A(t)xh(t)) + (A(t)xp(t) + b(t))

= (xh(t)) + (xp(t)) = (xs(t))

(4)

xs(t) (1) xs(t) (3)(1)xs(t0) = x0xh(t)(2)(2)

xh(t0) = x0 xp(t0) (5)

(2)

2014

A. A

sano

, Kan

sai U

niv.

t

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014 (2)

x = A(t)x+ b(t) (1)

xp(t) b(t) 0

x = A(t)x (2)

xh(t)(1)

xs(t) = xh(t) + xp(t) (3)

(3) xs(t) (1)(1)

A(t)xs(t) + b(t)

= A(t) (xh(t) + xp(t)) + b(t)

= (A(t)xh(t)) + (A(t)xp(t) + b(t))

= (xh(t)) + (xp(t)) = (xs(t))

(4)

xs(t) (1) xs(t) (3)(1)xs(t0) = x0xh(t)(2)(2)

xh(t0) = x0 xp(t0) (5)

(2)

2014

A. A

sano

, Kan

sai U

niv.

t

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

1. x(t0), x(t0)

2. 2 x1(t), x2(t)C1x1(t) + C2x2(t)C1, C2

2.C1x1(t) + C2x2(t) = 0 tC1 = C2 = 02.2

2014 (2)

x = A(t)x+ b(t) (1)

xp(t) b(t) 0

x = A(t)x (2)

xh(t)(1)

xs(t) = xh(t) + xp(t) (3)

(3) xs(t) (1)(1)

A(t)xs(t) + b(t)

= A(t) (xh(t) + xp(t)) + b(t)

= (A(t)xh(t)) + (A(t)xp(t) + b(t))

= (xh(t)) + (xp(t)) = (xs(t))

(4)

xs(t) (1) xs(t) (3)(1)xs(t0) = x0xh(t)(2)(2)

xh(t0) = x0 xp(t0) (5)

(2)

2014

A. A

sano

, Kan

sai U

niv.

t

2014 (1)

x(t)

x + P (t)x +Q(t)x = R(t) (1)

0P (t)Q(t)a, b

x + ax + bx = 0 (2)

x 0

2 + a+ b)et = 0

(3)

Education

2014年度秋学期 応用数学（解析） 第２部・基本的な微分方程式 ／ 第８回 ２階線形微分方程式（２） (2014. 11. 13)

2014年度秋学期　応用数学（解析）　第２部・基本的な微分方程式　／　第８回　２階線形微分方程式（２） (2014. 11. 13)