4.6: 3 sin 5 sin θ θ t θ 2t θ 4t : sin ωt ω sin θ θ ωt sin ωt 1 ω ω [rad/sec] 1 [sec] ω[rad] [rad/sec] 5.3 ω [rad/sec] 5.7: 2t 4t sin 2t sin 4t

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1 1 1.1 sin 2π [rad] 3 ft 3 sin 2t π : sin θ 2.2 sin θ ft t t [sec] t sin 2t π 4 [rad] sin sin θ θ t θ 2t π : 2.2: sin θ θ θ [rad] [rad] 4 sin θ sin 2t π 4 sin : : 1

2 4.6: 3 sin 5 sin θ θ t θ 2t θ 4t : sin ωt ω sin θ θ ωt sin ωt 1 ω ω [rad/sec] 1 [sec] ω[rad] [rad/sec] 5.3 ω [rad/sec] 5.7: 2t 4t sin 2t sin 4t sin 2t 1 2[rad] sin 4t 1 4[rad] sin 4t 2 sin ωt ω ω 5.9: ω ω ω 5.8 ω 2

3 : T 6.12: ω [rad/sec] T [sec] π T 1 T 2π : ft T ft + T ft 6.4 ft T 6.4 T T 2T, 3T T T 2T T 2T, 3T, T, 2T, 3T, 6.2 ω ω T 6.12 ω T ω T 6.13: 1 ω[rad/sec] ω [rad] 6.6 T ωt [rad] ω [rad/sec] T [sec] ωt 2π

4 7.14 [Hz] f ω[rad/sec] f[hz] ω 2πf ω, T, f : T 0.5[sec] 1 2 f 2[Hz] 7.15 ωt 2π 8.13 f 1 T 8.14 ω 2πf ω, T, f : 2 Hz T 2 [sec] f 0.5 [Hz] 7.2 T f T [sec] f[hz] f 1 T ω 2 sin ωt sin ωt π : sin ωt sin ωt π ω[rad/sec] f[hz] ω [rad/sec] ω [rad] 7.10 f[hz] 1 f 1 1 2π [rad] f 1 2πf [rad]

5 9.18: 9.19 sin ωt sin ωt π 2 π : 9.17: sin ωt π 2 sin ωt π π 2 sin ωt sin ωt π 2 π 2 π 2 sin ωt π 2 sinωt + ϕ 9.16 ϕ [rad] sin ωt sin ωt sin ωt sin ωt π 2 π 2 2 sin ϕ ϕ l ϕ l ϕ π π 2 ϕ l

6 1 [rad] π[rad] 1 T[sec] ϕ l ϕ 10.22: : ϕ l π 2 ϕ l 2π 1 ω : T, l ϕ l [rad] ϕ l : l 2π : T ϕ l : 1 2π : ϕ l π : ϕ l f 1 t f 2 t π 3 [rad]f 1t f 2 t π 3 [rad] 6

7 9.20 ω : T [sec] 2 l [sec] ϕ l [rad] ϕ l 2π T l [rad] π T ω ω [rad/sec] 2 l [sec] ϕ l [rad] ϕ l ωl [rad] : ω [rad/sec] 2 ϕ l l ϕ l ω [sec] [sec] ϕ l f 1 t f 2 t π 3 [rad] [rad]

8 t t 12.1: : : t ut yt t : y t + 4yt 10 sin 2t 13.1 y t yt t t 13.1 t 0 y 3 y0 3 y t + 4yt 10 sin 2t, y : 13.2 yt yt 8

9 13.2 y t + 4yt 10 sin 2t, y ut ut 10 sin 2t yt : yt : ut yt 13.7 y t + 4yt ut : 13.4 y t 4yt + ut 13.5 y t yt 13.5 y t t yt ut : yt y0 y0 3 yt : yt 13.11: y t + 4yt 10 sin 2t, y yt 9

10 y t + 4yt ut ut 10 sin 2t y0 3 t yt 14 y0 ut yt ut y0 t yt λ 4 yt Ce 4t C y y0 Ce C yt 3e 4t y t + ayt 0 a yt ayt : t yt yt yt yt e t t yt e λt λ y t λe λt λ λ λe λt + ae λt e λt λ + a λ a y t + 4yt 0, y λ λ yt e at λ λ a 10

11 15.14 yt C yt Ce at y t + ayt ace at + ace at y t + ayt yt Ce at C 15.8 y C C 3 y t + ayt 0, y0 b yt be at ut y : be at a ut 0 a a a y t+ayt 0 y t λ yt 1 λ + a y t yt 2 y t, yt t y t yt 1 y 2 t sin yt 0 yt : 11

12 ut 1 0 y 0 t 2 y p t 3 yt y 0 t + y p t 4 yt y t + 4yt 10 sin 2t, y yt 1 0 y 0t + 4y 0 t λ+4 0 λ 4 y 0 t Ce 4t C 2 y pt + 4y p t 10 sin 2t y p t 10 sin 2t y p t d 1 cos 2t + d 2 sin 2t d 1, d d 1 cos 2t + d 2 sin 2t +4d 1 cos 2t + d 2 sin 2t 10 sin 2t d 1 sin 2t + 2d 2 cos 2t +4d 1 cos 2t + d 2 sin 2t 10 sin 2t cos 2t sin 2t 2d 2 +4d 1 cos 2t+ 2d 1 +4d 2 sin 2t 10 sin 2t { 2d 2 + 4d 1 0 2d 1 + 4d d 1 1, d y p t cos 2t + 2 sin 2t y t + 4yt 10 sin 2t yt y 0 t + y p t Ce 4t cos 2t + 2 sin 2t y0 3 C t 0 3 C C yt 4e 4t cos 2t + 2 sin 2t y 0t + 4y 0 t y 0 t Ce 4t C y pt + 4y p t 10 sin 2t

13 17.37 y p t cos 2t + 2 sin 2t t 0 y p0 1 y y t + 4yt 10 sin 2t yt y 0 t + y p t y t + 4yt y 0 t + y p t + 4y 0 t + y p t y 0t + 4y 0 t + y pt + 4y p t sin 2t ut 10 sin 2t y p t cos 2t sin 2t sin 2t y p t cos 2t sin 2t d 1 d y p t ut ut ut ut k y p t d d ut k cos ωt ut k sin ωt y p t d 1 cos ωt + d 2 sin ωt d 1, d 2 ut ke γt y p t de γt d ut ke αt cos ωt ut ke αt sin ωt y p t e αt d 1 cos ωt+d 2 sin ωt d 1, d 2 ut kt n y p t d n t n + d n 1 t n 1 + d 1 t + d d n,, d : 13

14 y t + 4yt 10 sin 2t, y yt 4e 4t cos 2t + 2 sin 2t : 18.16: 1 4e 4t λ cos 2t + 2 sin 2t 2 yt 18.2 yt 4e 4t cos 2t + 2 sin 2t e 4t t 0 e at a t λ 4 λ 4 e 4t t y t + 2yt 3, y : e at 1 4e 4t yt

15 19.2 λ 2, y t + 3yt 2e 5t, y yt C 1 e 2t + C 2 e 3t C 1, C y t, y t, yt λ 2, λ, 1 1 e λt λ e λt + 5e λt + 6e λt λ 2 e λt + 5λe λt + 6e λt e λt λ 2 + 5λ λ e λt λ yt, y t 2 y t λ 2, e 2t e 3t C 1, C 2 C 1 e 2t C 2 e 3t C 1 e 2t + C 2 e 3t yt C 1 e 2t + C 2 e 3t C 1, C y t + 5y t + 6yt λ 2 + 5λ λ + 2λ y t + 5y t + 6yt 0, y0 1, y 0 1 yt

16 λ 2 + 5λ λ 2, y t + py t + qyt yt C 1 e 2t + C 2 e 3t C 1, C 2 y0 1 t C 1 + C y t 2C 1 e 2t 3C 2 e 3t y 0 1 2C 1 3C C 1 4, C yt 4e 2t + 3e 3t y0 1 y λ 2 + pλ + q λ 1, λ 2 2 λ yt C 1 e λ1t + C 2 e λ2t yt C 1 + C 2 te λt α ± βj yt e αt C 1 cos βt + C 2 sin βt C 1, C E. ; 21.5 y t + 2y t + 10yt : y t + 2y t + yt

17 微分方程式 解答は図 を参照 図 21.20: 微分方程式の解法 2 階の非斉次微分方程式 非斉次微分方程式の初期値問題 非斉次の 右辺が 0 でない 場合で 初期値が指定さ れている問題に対する解法は 第 17.1 節の方法と同じで ある 1 右辺を 0 に置き換えた斉次方程式の一般解 y0 t を求める 2 非斉次方程式の特殊解 yp t を求める 3 非斉次方程式の一般解 yt y0 t + yp t を求 める 4 一般解 yt に初期条件をあてはめて 任意だっ た定数の具体的な値を求める 2 で特殊解を求めるときは 第 17.3 節を参照すると よい 22.2 右辺が定数の場合 右辺が指数関数の場合 y t+2y t+10yt 30e 2t, y0 2, y 0 1 y t + 3y t + 2yt 6, y0 1, y 0 1 を満たす yt を求めよ 解答は図 を参照 図 22.21: 微分方程式の解法 を満たす yt を求めよ 解答は図 を参照 17 作成 山形大学工学部 村松

18 微分方程式 図 22.22: 微分方程式の解法 22.4 右辺が三角関数の場合 y t + 2y t + yt sin t, y0 0, y 0 1 を満たす yt を求めよ 微分方程式に関する基礎用語 線形と非線形 1 階の微分方程式のうち y t + ptyt rt という形に書けるものを線形微分方程式という この形以 外のものは非線形となる 例えば y t + yt t は線形な微分方程式である 一方 y t + yt y 2 t 作成 山形大学工学部 村松

19 23.88 t y 2 t y t + yt sin t y t + sin yt sin t sin yt yt C 1 e 2t + C 2 e 3t C 1 C 2 yt yt yty t + yt sin t y t yt 2 y t + pty t + qtyt rt y t + 3y t + 2yt e 4t y t + 3 sin ty t + t 2 yt e 4t pt qt y t, yt t 23.3 y t + ptyt rt y t + pty t + qtyt rt rt 0 rt 0 t rt 0 rt 0 rt y t + 5y t + 6yt

20 24 Ax b 24.1 A A : m n m n m n M m 11 m 12 m 13 m 14 m 21 m 22 m 23 m 24 m 31 m 32 m 33 m i j m ij m ij M i, j m 11 m 12 m 13 m 14 M m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 [ ] 24.2 v , v n n 24.4 v 1 v M, v a, k i, j, n 25 A B a b p A, B c d r q s 25.5 A B A + B a b p q a + p b + q A + B + c d r s c + r d + s

21 A B a b p A B c d r A q a p b q s c r d s , B 5 7 A B AB a b p q ap + br aq + bs AB c d r s cp + dr cq + ds A + B A + B A B A B A B 26 k A k k a b ka kb A ka 26.8 c d kc kd 27.2: A, B scalar multiplication 5 6 k 2, A ka ka AB AB A , B AB 27.1 A B a b A, B c d p r q s 27.9 AB

22 A , B A, B, C k AkB kab kab ABC ABC AB + C AB + AC A + BC AC + BC AB AB A B A B A m n B n l AB m l : n n n I n 27.3: 28.6: M, N MI M, IN N : 27.2 AB BA AB BA AB BA A, B ABC ABC E I 28.3 M M M T 1 4 M T

23 M i, j M T j, i T v v v 1 v 2. v n T v 1 v 2 v n v 1 v 2. T v 1 v 2 v n m m.... M m nn m k M k 0 m k m k nn k v n 28.4 M M T M M M i, j j, i : i, i : m 11 m 12 m nn m M mn 1,n 0 0 m nn m k 11??. M k 0 m k ? 0 0 m k nn ? k m m M 21 m m n1 m n,n 1 m nn m k M k? m k ?? m k nn : 29.1 M MM 1 M 1 M I

24 M 1 M M M M M a b M c d M 1 M 1 1 ad bc 2 2 d c b a ad bc : A 1 30 AB T B T A T, ABC T C T B T A T M M M a, x, b ax b a 0 x b a A, X, B AX B A X A 1 B X B A X BA A 1, B 1, C 1 AB 1 B 1 A 1, ABC 1 C 1 B 1 A 1 M M M T 1 M 1 T M a c b d M 2 a + dm + ad bci n n 2 II 32 24

25 v 1, v v xy x y 32.13: v k v kv 32.11: v , v xyz x y z : v 1 v 2 2 v 1 + 3v v 1 3v : v : v v v 1 3v 2 v 1 3v 2 v 1 3v v 1 + 3v 2 25

26 32.5 v 1, v 2 v a v b v 1 + 3v v 1, v : 32.16: v 1 v 2 2 m v 1, v 2,, v m k 1, k 2,, k m M v a v b M k 1 v 1 + k 2 v k m v m v 1, v 2,, v m b k 1 v 1 + k 2 v k m v m b k 1, k 2,, k m : v a M v b Mv a v b : 26

27 33.19 B A AB : 2 xy O A v a v ax v ay θ v b v bx v by v a v b M v b Mv a v a r x α vax r cos α v a v ay r sin α v b θ α + θ vbx r cosα + θ v b v by r sinα + θ v b r cosα + θ v b r sinα + θ rcos α cos θ sin α sin θ rsin α cos θ + cos α sin θ cos θ r cos α sin θ r sin α sin θ r cos α + cos θ r sin α cos θ sin θ r cos α sin θ cos θ r sin α cos θ sin θ vax sin θ cos θ v ay xyz 3 M cos θ sin θ M sin θ cos θ v bx cos θ sin θ v ax sin θ cos θ v by v ay : x x θ v bx v by v bz cos θ sin θ 0 sin θ cos θ v ax cos θv ay sin θv az v ax v ay v az sin θv ay + cos θv az 34.21: x x v bx v ax y z 27

28 x yz θ sin θ, cos θ 1 A B P P 34.25: : x yz M x 0 cos θ sin θ sin θ cos θ x y cos θ 0 sin θ M y sin θ 0 cos θ z cos θ sin θ 0 M z sin θ cos θ y 2 sin θ y xz x z θ y z θ P x 3, y 3 x 1, y v : y xz 28

29 : v v v 1 2 x y xy 0, 3 2 y 2x : 35.27: V v V V 1 v 1 v V V v v 1 2 V x y 36.29: 29

30 , 3 p, a, v t v v p + ta ta t p + ta v p a xyz 1, 2, 3 T a T v x y z T 1, 2, 3 p a ta t v p + ta x 1 y 2 + t z : x 0 v, p y x, y x, y a 1 a x 1 y x t y t t x y y 2x t t x 1 3 y z v w v, w v w v w v, w n 30

31 37.1 v + w, x v, x + w, x v, w + x v, w + v, x kv, w v, kw kv, w v, w w, v v, v 0 v, v 0 v v w v, w v w v v v, v v v v 37.4 v, w v, w v w Schwarz v + w v + w t 0 tv + w 2 tv + w, tv + w v 2 t 2 + 2v, wt + w 2 v 2 t 2 + 2v, wt + w v, w 2 4 v 2 w v + w 2 v 2 + 2v, w + w 2 v v, w + w v v w + w v + w v v 1 v 2, w w 1 w v, w v 1 w 1 + v 2 w v v 1 v 2 v 3, w w 1 w 2 w v, w v 1 w 1 + v 2 w 2 + v 3 w v, w v T w n v a 1 a 2. a n b 1, w b 2. b n v, w a 1 b 1 + a 2 b a n b n v T w v v 2 v v 1 v 2 v v, v v v

32 v 1 v v v k a, b kb, b b, b b 2 k k c a, b b a, b b 2 b : 2 v 1 v 2 v 1 v : a b : a b θ a b θ cos θ a, b a b θ π cos θ θ : a b a b a b c a, b c a, b b 2 b c a b a b a b c b k c kb k a, b kb a kb b a kb, b : a a 1 e a e a a a a 2 a 2 1 a/ a 1 32

33 38 xyz P p x, p y, p z a a x a y a z T v p T a x p x y p y T a x a y z p z a z a x x + a y y + a z z p x a x p y a y p z a z xyz P p x, p y, p z a a x a y a z T 38.36: P a P p p x p y p z x, y, z v x y z xyz x, y, z v x, y, z a x x + a y y + a z z + d d p x a x p y a y p z a z xyz a a a x, a y, a z x, y, z : a b a b 38.37: P a v x, y, z a v p a v p, a a, b θ 0 θ π a b a b a b a b a b 2 a b 33

34 a b a b a b a b b a a a a b a b sin θ ka b a kb ka b a b b ka a b a b 0 3 a, b, c a b, c b c, a c a, b a, b, c 6 a b a y b z a z b y a b a z b x a x b z a x b y a y b x a b a b 1 4 a 2, b a b a b a b + c a b + a c a + b c a c + b c a b c a b c : a b a b xyz a a x a y, b b x b y a z b z 34

35 M detm M detdeterminant a M c a M c M b d b d 40.1 ad bc 40.2 m 11 m 12 m 13 m 21 m 22 m 23 m 31 m 32 m m 11 m 12 m 13 M m 21 m 22 m 23 m 31 m 32 m 33 m 22 m 23 m 11 m 32 m 33 m m 12 m m 32 m 33 m 12 m 13 +m 31 m 22 m 23 m 13 m 21 m 32 + m 11 m 22 m 33 + m 12 m 23 m 31 m 12 m 21 m 33 m 13 m 22 m 31 m 11 m 23 m m 11 m 12 m 13 m 14 M m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 m 41 m 42 m 43 m m 11 m 12 m 13 m 14 M m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 m 41 m 42 m 43 m 44 m 22 m 23 m 24 m 11 m 32 m 33 m 34 m 42 m 43 m 44 m 12 m 13 m 14 m 21 m 32 m 33 m 34 m 42 m 43 m 44 m 12 m 13 m 14 +m 31 m 22 m 23 m 24 m 42 m 43 m 44 m 12 m 13 m 14 m 41 m 22 m 23 m 24 m 32 m 33 m M 1 M 2 n M 1 M 2 M 1 M I I M M T M M m m M m nn 35

36 M m 11 m 22 m nn M m 11 m 12 m nn. 0 m M m n 1,n 0 0 m nn m m 21 m M m n1 m n,n 1 m nn M m 11 m 22 m nn M 11, M 22 M 11 M 12 M M 22 M M 11 M M 11, M 22 M 11 0 M M 21 M M M 11 M A A 0 A A 0 A A A 1 2 A A A A A 41.2 A A A A A 1 1 A A A 1 A 0 A 1 A 41.3 A n n A A A A 1 Ax b x A 1 b 41.4 T n n x z n x z T x z 36

37 41.1: x T 1 z x z T z T x x z x n z n x 0 z 0 z 0 x 0 xt 0 zt T xt zt a 2. b x 1 + 4x 2 + 7x x 1 + 5x 2 + 8x x 1 + 6x 2 + 9x 3 12 x 1 x 2 x x 1 10 A 2 5 8, x, b x 2 x Ax b Ax b M A b M c 1. d 2. e 1 0. f 1. a, b, c d,e,f x y x, y, z M z

38 M c a b c c3 x 3, y 3, z 1 c a1 2 3 b c c A M [A I] I M A M M M A : 2 x y y 1 1 y x A A A 1 M 38

39 i j i j j 45.2 z z a + bj 45.1 a, b a b : 1 z z a + jb re jθ a, b, r, θ 45.4: a, b, r, θ r a 2 + b a cos θ a2 + b, sin θ b 2 a2 + b : tan θ b a, θ tan 1 b a 45.6 z z re jθ 45.2 r θ e jθ θ e jθ cos θ + j sin θ 45.7 e jθ θ θ e jθ : r z z θ z : e jθ 39

40 e jθ e jθ θ e jθ θ e jθ θ π 2 ejθ 45.6 z 1 r 1 e jθ 1, z 2 r 2 e jθ z 1 z 2 r 1 e jθ1 r 2 e jθ 2 r 1 r 2 e jθ1+θ z 1, z 2 z 1 z e j π 2 j : : θ π 2 ejθ j e j π 2 j, e j0 1, e jπ z 1 a + jb, z 2 c + jd z 1 + z 2 a + c + jb + d z 1 z 2 z 1 z 2, z 1 z 2 z 1 + z z1 z 2 z 1 a + jb z 2 c + jd a + jbc jd c + jdc jd ac + bd + jbc ad c 2 + d 2 ac + bd bc ad c 2 + j + d2 c 2 + d z 1 z 2 r 1e jθ1 r 2 e jθ 2 r 1 r 2 e jθ 1 θ : z 1 z 2 j 2 1 z 1 z 2 a + jbc + jd ac + jad + jbc + j 2 bd ac bd + jad + bc z 1 z 2 z 1 z 2, z 1 z 2 z 1 z z z 2, 1 z z 2 z 2 z n z n re jθ n r n e jnθ r n cos nθ + j sin nθ

41 z n n z n r 1 θ π 2 e j π 2 j 1 j cos θ + j sin θ n cos nθ + j sin nθ e j π 2 j j π/2 π 2 π : j π : z 1 θ 2 z 2 θ 2 z 1 e jθ θ z a + jb z a jb : 45.10: e jθ θ θ π e j π 2 j j π/ z re jθ z re jθ z 1 + z 2 z 1 + z 2, z 1 z 2 z 1 z z 1 z 2 z 1 z 2, z1 z 2 z 1 z zz z z + z 2 z z z 2j z : j π 2 41

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