.. p.2/5
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- つづる おうじ
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1 IV. p./5
2 .. p.2/5
3 .. 8 >< >: d dt y = a, y + a,2 y a,n y n + f (t) d dt y 2 = a 2, y + a 2,2 y a 2,n y n + f 2 (t). d dt y n = a n, y + a n,2 y a n,n y n + f n (t) (a i,j ) p.2/5
4 .. 8 >< >: d dt y = a, y + a,2 y a,n y n + f (t) d dt y 2 = a 2, y + a 2,2 y a 2,n y n + f 2 (t). d dt y n = a n, y + a n,2 y a n,n y n + f n (t) (a i,j ) y(t) = (y (t), y 2 (t),..., y n (t)) p.2/5
5 .. 8 >< >: d dt y = a, y + a,2 y a,n y n + f (t) d dt y 2 = a 2, y + a 2,2 y a 2,n y n + f 2 (t). d dt y n = a n, y + a n,2 y a n,n y n + f n (t) (a i,j ) y(t) = (y (t), y 2 (t),..., y n (t)) f i (t) ( ), (i =,..., n) 8 >< >: d dt y = a, y + a,2 y a,n y n d dt y 2 = a 2, y + a 2,2 y a 2,n y n. d dt y n = a n, y + a n,2 y a n,n y n () ( ) p.2/5
6 .. y = y y 2. y n, A = C A a, a,2 a,n a 2, a 2,2 a 2,n C A a n, a n,2 a n,n p.3/5
7 .. y = y y 2. y n, A = C A a, a,2 a,n a 2, a 2,2 a 2,n C A a n, a n,2 a n,n () = Ay (2) dt p.3/5
8 .. y = y y 2. y n, A = C A a, a,2 a,n a 2, a 2,2 a 2,n C A a n, a n,2 a n,n () = Ay (2) (2) n y (t),y 2 (t),...,y n (t) dt W(y,y 2,...,y n )(t) = (y (t) y 2 (t) y n (t)) p.3/5
9 .. y = y y 2. y n, A = C A a, a,2 a,n a 2, a 2,2 a 2,n C A a n, a n,2 a n,n () = Ay (2) (2) n y (t),y 2 (t),...,y n (t) dt W(y,y 2,...,y n )(t) = (y (t) y 2 (t) y n (t)) n (2) p.3/5
10 .. p.4/5
11 .. [ ] p.4/5
12 .. [ ] () A {λ,..., λ n } λ i (i =,..., n) u i (2) y(t) = e λ it u i p.4/5
13 .. [ ] () A {λ,..., λ n } λ i (i =,..., n) u i y(t) = e λ it u i (2) (2) {u,u 2,...,u n } (2) y(t) = c e λ t u + c 2 e λ 2t u c n e λ nt u n (c, c 2,..., c n ) p.4/5
14 .. 8 >< >: d dt y = y + y 2 + y 3 d dt y 2 = y y 2 d dt y 3 = y y 3 i.e. dt = C A y p.5/5
15 .. 8 >< >: d dt y = y + y 2 + y 3 d dt y 2 = y y 2 d dt y 3 = y y 3 i.e. dt = C A y [ ] ( ± i) C A, ±i C A, C A p.5/5
16 .. 8 >< >: d dt y = y + y 2 + y 3 d dt y 2 = y y 2 d dt y 3 = y y 3 i.e. dt = C A y [ ] ( ± i) C A, ±i C A, C A ( + i) ( i) y(t) = c e t C A + c 2e it C A + c 3e it C A p.5/5
17 .. 8 >< >: d dt y = y + y 2 + y 3 d dt y 2 = y y 2 d dt y 3 = y y 3 i.e. dt = C A y [ ] ( ± i) C A, ±i C A, C A ( + i) ( i) y(t) = c e t C A + c 2e it C A + c 3e it C A c = c, c 2 = c 2 + c 2, c 3 = i(c 2 c 3 ) p.5/5
18 .. 8 >< >: d dt y = y + y 2 + y 3 d dt y 2 = y y 2 d dt y 3 = y y 3 i.e. dt = C A y [ ] ( ± i) C A, ±i C A, C A ( + i) ( i) y(t) = c e t C A + c 2e it C A + c 3e it C A c = c, c 2 = c 2 + c 2, c 3 = i(c 2 c 3 ) sin t cos t sin t cos t y(t) = c e t C A + c B cos t C A + c B sin t C A cos t sin t p.5/5
19 .. [ ] () dt 6 3 Ay, (2) 2 dt 3 2 Ay p.6/5
20 .. [ ] () dt 6 3 Ay, (2) 2 dt 3 2 Ay [ ] () y(t) = c e 3 A + c 2 e A 2 p.6/5
21 .. [ ] () dt 6 3 Ay, (2) 2 dt 3 2 Ay [ ] () y(t) = c e 3 A + c 2 e A 2 (2) y(t) = c e ( 2 A + c 2 e ( 2 2 A + i i p.6/5
22 .. [ ] () dt 6 3 Ay, (2) 2 dt 3 2 Ay [ ] () y(t) = c e 3 A + c 2 e A 2 (2) y(t) = c e ( 2 A + c 2 e ( 2 2 A + i i c = c + c 2, c 2 = i(c c 2 ) p.6/5
23 .. [ ] () dt 6 3 Ay, (2) 2 dt 3 2 Ay [ ] () y(t) = c e 3 A + c 2 e A 2 (2) y(t) = c e ( 2 A + c 2 e ( 2 2 A + i i c = c + c 2, c 2 = i(c c 2 ) y(t) = c e 2cos t A + c 2e 2sin t A cos t sin t cos t + sin t p.6/5
24 ..2 p.7/5
25 ..2 A A e ta (= exp ta) = E + ta + t2 2 A2 + + tn n! An + = X n= n! (ta)n p.7/5
26 ..2 A A e ta (= exp ta) = E + ta + t2 2 A2 + + tn n! An + = e A = E, e sa e ta = e (s+t)a, e ta = X n= ne tao n! (ta)n p.7/5
27 ..2 A A e ta (= exp ta) = E + ta + t2 2 A2 + + tn n! An + = e A = E, e sa e ta = e (s+t)a, e ta = X n= ne tao n! (ta)n d dt eta = Ae ta = e ta A p.7/5
28 ..2 A A e ta (= exp ta) = E + ta + t2 2 A2 + + tn n! An + = e A = E, e sa e ta = e (s+t)a, e ta = X n= ne tao n! (ta)n d dt eta = Ae ta = e ta A = Ay (3) dt c = (c, c 2,..., c n ) y(t) = e ta c p.7/5
29 ..2 A A e ta (= exp ta) = E + ta + t2 2 A2 + + tn n! An + = e A = E, e sa e ta = e (s+t)a, e ta = X n= ne tao n! (ta)n d dt eta = Ae ta = e ta A = Ay (3) dt c = (c, c 2,..., c n ) y(t) = e ta c e ta (3) ( ) p.7/5
30 ..2 dt Ay p.8/5
31 ..2 dt Ay [ ] exp A = E + A + t2 A tn A n + p.8/5
32 ..2 dt Ay [ ] exp A = E + A + t2 2 = E + A + t2 2 A + + n A + 2 A + + n A + n! p.8/5
33 ..2 dt Ay [ ] exp A = E + A + t2 2 = E + A + t2 2 P t n P t n n! = Bn= P n= (n )! t n 2 A + + n A + 2 A + + n A + n! C A p.8/5
34 ..2 dt Ay [ ] exp A = E + A + t2 2 = E + A + t2 2 P t n P t n n! = Bn= P n= (n )! t n 2 A + + n A + 2 A + + n A + n! C A et te t A e t p.8/5
35 ..2 dt Ay [ ] exp A = E + A + t2 2 = E + A + t2 2 P t n P t n n! = Bn= P n= (n )! y(t) = et A + c tet A e t t n 2 A + + n A + 2 A + + n A + n! C A et te t A e t p.8/5
36 ..2 A J P A = PJP p.9/5
37 ..2 A J P A = PJP A n = (PJP ) n = PJP PJP PJP = PJ n P p.9/5
38 ..2 A J P A = PJP A n = (PJP ) n = PJP PJP PJP = PJ n P A e ta = E+tPJP + t2 2 PJ2 P + + tn n! PJn P + = P X n=! n! (tj)n P = Pe tj P p.9/5
39 ..2 A J P A = PJP A n = (PJP ) n = PJP PJP PJP = PJ n P A! X e ta = E+tPJP + t2 2 PJ2 P + + tn n! PJn P + = P n! (tj)n P = Pe tj P n= (3) y(t) = e ta c = Pe tj P c p.9/5
40 ..2 A J P A = PJP A n = (PJP ) n = PJP PJP PJP = PJ n P A! X e ta = E+tPJP + t2 2 PJ2 P + + tn n! PJn P + = P n! (tj)n P = Pe tj P n= (3) y(t) = e ta c = Pe tj P c [ ] dt Ay p.9/5
41 ..2 A J P A = PJP A n = (PJP ) n = PJP PJP PJP = PJ n P A! X e ta = E+tPJP + t2 2 PJ2 P + + tn n! PJn P + = P n! (tj)n P = Pe tj P n= (3) y(t) = e ta c = Pe tj P c [ ] dt Ay [ 2 A A p.9/5
42 ..2 A J P A = PJP A n = (PJP ) n = PJP PJP PJP = PJ n P A! X e ta = E+tPJP + t2 2 PJ2 P + + tn n! PJn P + = P n! (tj)n P = Pe tj P n= (3) y(t) = e ta c = Pe tj P c [ ] dt Ay [ 2 A A@ 2 A@ 2 A@ c A e 2t c A p.9/5
43 ..2 A J P A = PJP A n = (PJP ) n = PJP PJP PJP = PJ n P A! X e ta = E+tPJP + t2 2 PJ2 P + + tn n! PJn P + = P n! (tj)n P = Pe tj P n= (3) y(t) = e ta c = Pe tj P c [ ] dt Ay [ 2 A A A@ 2 A@ 2 A@ c A = e2t A+c 2 2 e2t A e 2t c e2t e2t p.9/5
44 ..3 p./5
45 ..3 [ ] p./5
46 ..3 [ ] = Ay + f(t) (4) dt p./5
47 ..3 [ ] = Ay + f(t) (4) dt «e ta dt Ay = d e ta y (4) dt p./5
48 ..3 [ ] = Ay + f(t) (4) dt «e ta dt Ay = d e ta y (4) dt d dt e ta y = e ta f(t) p./5
49 ..3 [ ] = Ay + f(t) (4) dt «e ta dt Ay = d e ta y (4) dt d dt e ta y = e ta f(t) e ta y = Z t t e τa f(τ)dτ + c, c p./5
50 ..3 [ ] = Ay + f(t) (4) dt «e ta dt Ay = d e ta y (4) dt d dt e ta y = e ta f(t) e ta y = Z t t e τa f(τ)dτ + c, c y = e ta c + Z t t e (t τ)a f(τ)dτ p./5
51 ..3 dt 2 Ay + e2t 2e 2t A p./5
52 ..3 dt 2 Ay + e2t 2e 2t A [ ] exp t@ 2 A=e cos t 2 sin t sin t A, cos t 8 < exp : τ 2 = A e2τ 2e 2τ cos τ + 2 sin τ A sin τ + 2cos τ p./5
53 ..3 dt 2 Ay + e2t 2e 2t A [ ] exp t@ 2 A=e cos t 2 sin t sin t A, cos t 8 < exp : τ 2 = A e2τ 2e 2τ cos τ + 2 sin τ A sin τ + 2cos τ y = exp 2 c A+exp 2 A 2 2 c 2 Z t 8 < exp : τ 2 = e2τ A dτ ; 2e 2τ p./5
54 ..3 dt 2 Ay + e2t 2e 2t A [ ] exp t@ 2 A=e cos t 2 sin t sin t A, cos t 8 < exp : τ 2 = A e2τ 2e 2τ cos τ + 2 sin τ A sin τ + 2cos τ y = exp 2 c A+exp 2 A 2 c 2 2 = e cos t sin t c A + e cos t sin t cos t sin t c < exp : 2 = A ; sin t sin t 2 cos t + 2 A cos t cos t + 2 sin t Z e2τ 2e 2τ A dτ p./5
55 ..3 dt 2 Ay + e2t 2e 2t A [ ] exp t@ 2 A=e cos t 2 sin t sin t A, cos t 8 < exp : τ 2 = A e2τ 2e 2τ cos τ + 2 sin τ A sin τ + 2cos τ 8 9 y = exp 2 c A+exp 2 Z t < A exp 2 c 2 2 : 2 = A ; = e cos t sin t c A + e cos t sin t sin t 2 cos t + 2 A sin t cos t c 2 sin t cos t cos t + 2 sin t = e cos t sin t c A + e 2 A (c = c + 2, c 2 = c 2 ) sin t cos t c e2τ 2e 2τ A dτ p./5
56 ..3 [ ] p.2/5
57 ..3 [ ] dt 2 Ay+ e2t 2e 2t 8 < (D 2)y A + y 2 = e 2t : y + (D 2)y 2 = 2e 2t D = d «dt p.2/5
58 ..3 [ ] dt 2 Ay+ e2t 2e 2t 8 < (D 2)y A + y 2 = e 2t : y + (D 2)y 2 = 2e 2t D = d «dt [ ] (D 2) (D 2) 2 y + (D 2)y 2 = (D 2)e 2t = p.2/5
59 ..3 [ ] dt 2 Ay+ e2t 2e 2t 8 < (D 2)y A + y 2 = e 2t : y + (D 2)y 2 = 2e 2t D = d «dt [ ] (D 2) (D 2) 2 y + (D 2)y 2 = (D 2)e 2t = (D 2 4D + 5)y = 2e 2t p.2/5
60 ..3 [ ] dt 2 Ay+ e2t 2e 2t 8 < (D 2)y A + y 2 = e 2t : y + (D 2)y 2 = 2e 2t D = d «dt [ ] (D 2) (D 2) 2 y + (D 2)y 2 = (D 2)e 2t = (D 2 4D + 5)y = 2e 2t y = e 2t (c cos t + c 2 sin t) 2e 2t p.2/5
61 ..3 [ ] dt 2 Ay+ e2t 2e 2t 8 < (D 2)y A + y 2 = e 2t : y + (D 2)y 2 = 2e 2t D = d «dt [ ] (D 2) (D 2) 2 y + (D 2)y 2 = (D 2)e 2t = (D 2 4D + 5)y = 2e 2t y = e 2t (c cos t + c 2 sin t) 2e 2t y 2 = e 2t (D 2)y = e 2t (c sin t c 2 cos t) + e 2t p.2/5
62 ..3 [ ] dt 2 Ay+ e2t 2e 2t 8 < (D 2)y A + y 2 = e 2t : y + (D 2)y 2 = 2e 2t D = d «dt [ ] (D 2) (D 2) 2 y + (D 2)y 2 = (D 2)e 2t = (D 2 4D + 5)y = 2e 2t y = e 2t (c cos t + c 2 sin t) 2e 2t y 2 = e 2t (D 2)y = e 2t (c sin t c 2 cos t) + e 2t y = e cos t sin t c A + e 2 A sin t cos t c 2 p.2/5
63 ..3 [ ] 8 < (D + )y + (D + 3)y 2 = t D = d «:(D )y + (2D 2)y 2 = 2t, dt p.3/5
64 ..3 [ ] 8 < (D + )y + (D + 3)y 2 = t D = d «:(D )y + (2D 2)y 2 = 2t, dt [ ] (D + ) (D (D ) 2 y 2 = + 3t p.3/5
65 ..3 [ ] 8 < (D + )y + (D + 3)y 2 = t D = d «:(D )y + (2D 2)y 2 = 2t, dt [ ] (D + ) (D (D ) 2 y 2 = + 3t y 2 = c e t + c 2 te t + 3t + 7 p.3/5
66 ..3 [ ] 8 < (D + )y + (D + 3)y 2 = t D = d «:(D )y + (2D 2)y 2 = 2t, dt [ ] (D + ) (D (D ) 2 y 2 = + 3t y 2 = c e t + c 2 te t + 3t + 7 y 2 (D )y = 2c 2 e t + 8t + 8 p.3/5
67 ..3 [ ] 8 < (D + )y + (D + 3)y 2 = t D = d «:(D )y + (2D 2)y 2 = 2t, dt [ ] (D + ) (D (D ) 2 y 2 = + 3t y 2 = c e t + c 2 te t + 3t + 7 y 2 (D )y = 2c 2 e t + 8t + 8 y = ce t 2c 2 te t 8t 6 p.3/5
68 ..3 [ ] 8 < (D + )y + (D + 3)y 2 = t D = d «:(D )y + (2D 2)y 2 = 2t, dt [ ] (D + ) (D (D ) 2 y 2 = + 3t y 2 = c e t + c 2 te t + 3t + 7 y 2 (D )y = 2c 2 e t + 8t + 8 y = ce t 2c 2 te t 8t 6 y c «y = 2 c 2 2c e t 2c 2 te t 8t 6 p.3/5
69 2. p.4/5
70 2. 8 < : d 2 y dt 2 y + dz dt z = e t 2 dt 2y + d2 z dt 2 z=, y() =, y () =, z() =, z () = p.4/5
71 2. 8 < : d 2 y dt 2 y + dz dt z = e t 2 dt 2y + d2 z dt 2 z=, y() =, y () =, z() =, z () = [ ] 8 < (s 2 )Y (s) + (s )Z(s) = s+ Y = L(y), Z = L(z) : 2(s + )Y (s) + (s 2 )Z(s) s=, p.4/5
72 2. 8 < : d 2 y dt 2 y + dz dt z = e t 2 dt 2y + d2 z dt 2 z=, y() =, y () =, z() =, z () = [ ] 8 < (s 2 )Y (s) + (s )Z(s) = s+ Y = L(y), Z = L(z) : 2(s + )Y (s) + (s 2 )Z(s) s=, Y, Z 8 < Y (s)= : Z(s)= 2 s+ s s 2 + s 2 s+ + s s 2 + s 2 + p.4/5
73 2. 8 < : d 2 y dt 2 y + dz dt z = e t 2 dt 2y + d2 z dt 2 z=, y() =, y () =, z() =, z () = [ ] 8 < (s 2 )Y (s) + (s )Z(s) = s+ Y = L(y), Z = L(z) : 2(s + )Y (s) + (s 2 )Z(s) s=, Y, Z 8 < Y (s)= : Z(s)= 2 s+ s s 2 + s 2 s+ + s s 2 + s 2 + y(t) = e t cos t, z(t) = 2 et 2 e t + cos t sin t p.4/5
74 2. [ ] 8 < y y + z = y() = 2, z() = : 5y + z 3z=, p.5/5
75 2. [ ] 8 < y y + z = y() = 2, z() = : 5y + z 3z=, 8 < y(t)= 2 e2t (4 cos2t 3 sin2t) : z(t)= 2 e2t (2 cos2t + sin2t) p.5/5
数学演習：微分方程式
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[1.1] r 1 =10e j(ωt+π/4), r 2 =5e j(ωt+π/3), r 3 =3e j(ωt+π/6) ~r = ~r 1 + ~r 2 + ~r 3 = re j(ωt+φ) =(10e π 4 j +5e π 3 j +3e π 6 j )e jωt
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3.4.7 [.] =e j(t+/4), =5e j(t+/3), 3 =3e j(t+/6) ~ = ~ + ~ + ~ 3 = e j(t+φ) =(e 4 j +5e 3 j +3e 6 j )e jt = e jφ e jt cos φ =cos 4 +5cos 3 +3cos 6 =.69 sin φ =sin 4 +5sin 3 +3sin 6 =.9 =.69 +.9 =7.74 [.]
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