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1 K E N Z OU x 2x x + x 2. x 2 2x 2, 2 2 d 2 x x 2... d 2 x x d 2 x 2 + P t + P 2tx Qx x x, x x 2 P 2 tx P tx 2 + Qx x, x 2. d x 4 2 x 2 x x 2.3

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3 P P P P E 2 ««a b A A d b c d ad bc c a A D P AP 0 3 A P.4 P AP.4 P P P AP P x.5 P.5 d P x DP x.6 y P x x P y.6 dy Dy.7 d y 2 0 y 0 3 y 2 y 2... dy 2y y y 0e 2t y 2 dy 2 3y y 2 y 2 0e 3t,y y e 2t 0 y 0 y 0 e 3t y 2 0 e 2t 0 0 e 3t y0 x P y, y P x y0 P x0 e 2t 0 e 2t 0 x P y P y0 P P x0 0 e 3t 0 e 3t 2 e 2t 0 2 x0 0 e 3t x e 2t + 2e 3t e 2t + e 3t x 0 2e 2t 2e 3t 2e 2t e 3t x 2 0 x 2... x { x 0 + 2x 2 0}e 2t + 2{x 0 x 2 0}e 3t c e 2t + 2c 2 e 3t x 2 { x 0 + 2x 2 0}e 2t + {x 0 x 2 0}e 3t c e 2t + c 2 e 3t 3

4 2 2 «a b A c d a λ c b d λ λ2 a + dλ + ad bc 0 2 OK 0 a + d 2 4ad bc a d 2 + 4bc λ a + d/2.2 Ax.9 x A d ln x A x x0eat.0 e At A exponential matrix A e A e A E + A + 2! A2 + 3! A3 + + n! An + n0 n! An. A A a 0 A 0 b e A E + A + 2! A2 + 3! A a a 0 + a b 2! 0 b 3! 0 b + a + 2! a2 + 3! a b + 2! b2 + 3! b3 + e a 0 0 e b 3 + 4

5 b B 0 b e B b + + b + b b 2! 0 b 3! 0 b 0 b + + b 2 2b + b 3 3b b 2! 0 b 2 3! 0 b b + 2! b2 + 3! a ! b + 3 3! b b + 2! b2 + 3! b3 + e b e b e b 0 e b 0 0 α C α 0 e C α α + 0 α 0 α 0 2! α 0 3! α α + + α α 3 0 α 0 2! 0 α 2 3! α ! α2 + 4! α4 + c 3! α3 + 5! α5 + α + 3! α3 5! α5 + 2! α2 + 4! α4 + cos α sin α sin α cos α e C cos α 0 0 sin α cos α + 0 cos α sin α K C Kα e C 0 sin α e C e Kα e C e Kα E cos α + K sin α.2 e iθ cos α + i sin α 2 K 2 K 2 0 E 0 K.2 a b D D b a 0 0 D a + b Ea + Kb

6 e D e D e Ea+Kb e Ea e Kb e Ea cos b sin b e a cos b sin b sin b cos b sin b cos b D.3 a + bi e A ABBA e A+B e A e B A B AB BA 2 e P AP P e A P 3 e A e A 4 d eta Ae ta.3 e ta 2 D P AP e ta e tp DP P e td P.4 a A a 0 e ta e at 0 0 b 0 e bt a b A e ta e at t 0 a 0 a b c A e ta e at cos at b a sin bt sin bt cos bt 4 A e ta 0 4 6

7 2 e ta e 4t 80 >< B >: 4t 0 e 0 4t e e 4t t 0 t 0 0 C B 0 t > C A 0 B 0 4t@ >; 0 e C A e 0 B t 0 0 C A 2 A e ta A λ A λe 0 A λe λ 4λ 5 0,...λ 4, λ 2 5 p λ 4 p Ap λ p p p { 3p + 4p 2p λ 2 5 { P p, P 2 2 A D D P AP A P DP 2 a e ta e tp DP P e td P e 4t 0 2 2e 4t e 5t 2 0 e 5t 2e 4t 2e 5t A e ta 9 9 A λ A λe λ ,...λ 3, 3 e 4t + e 5t e 4t + 2e 5t 2 A 3 P D P AP

8 P P p, q.5 Ap, Aq A p, q AP P D 3 p, q 0 3 3p p 3q 3 3 3p, p-3q { Ap 3p Aq p 3q 3 4 p p 3 6p 4, 9p p q 2 q 3 6q 4 2, 9q q P P e ta e tp DP P e td P.7 b D P 3 AP 0 3 e td e 3t t A e ta P e td P e 3t A λ 2 t e 3t + 6t 4t 9t 6t e ta A λe λ ,...λ 2 + 3i, λ 2 λ 2 3i x 2 λ 2 + 3i p Ap λ p y { 5x 3y 2 + 3ix 6x y 2 + 3iy 0 p + i p + iq i 8

9 P p, q 0 P p, q, P 0 D P AP c e td e 2t cos 3t sin 3t sin 3t cos 3t e ta P e td P e 2t e 2t 0 cos 3t sin 3t sin 3t cos 3t cos 3t + sin 3t sin 3t 2 sin 3t cos 3t sin 3t 0 «a b A λ λ c d fλ λ 2 a + dλ + ad bc 0 «3 A 2 6 fλ λ 2 9λ + 20 λ 4λ 5 0,... λ 4, a b c d e f A g h i fλ λ 3 a + e + iλ 2 + ae bd + ai cg + ei fhλ + aei + bfg + cdh ceg bdi afh 0 2 Ax 2. c x e ta c, c c c 2 A λ, λ λ 2 λ 2 x, x2 2.2 x c e λt x + c 2 e λ2t x

10 2. 2 d x 7 2 x 7 2, A y 6 y 6 A λ 2 3λ ,...λ 5, 8 P Ax λx P 2, P /3 2/3 /3 /3 A D P 5 0 AP 0 8 λ 0 0 λ 2 x e ta c e P tdp c P e td P c P c c x 2 e 5t 0 y 0 e 8t c c 2 c e 5t + c 2 e 8 t 2 d x a b y c d x y, A a c b d 2.4 A λ, λ 2 Ax λx { a b x x ax + bx 2 λ x λ c d x 2 x 2 cx + 2 λ x 2 x b, x 2 λ a { a b x x ax + bx 2 λ 2 x λ 2 c d x 2 x 2 cx + 2 λ 2 x 2 x b, x 2 λ 2 a b b P λ a λ 2 a x P e td b b e λ t 0 c c λ a λ 2 a 0 e λ 2t.. x. c y b e λt + c 2 λ a b λ 2 a e λ 2t c A 2 p λ, λ 2,, λ λ 2 x y c p e λ t + c 2 q q e λ 2t 2.6 0

11 x b c e λt b + c 2 y λ a λ 2 a e λ 2t A 2 9 2x 6y dy 2x + 9y λ 2 λ + 30 λ 5λ 6 0, λ 5, 6 P Ax λx P x 2 c e 5t + c 2 y 3 2 e 6t 2.2 0x 9y dy 6x + 4y 0 9 A 6 4 λ 2 4λ + 4 λ 2 2 0, λ 2, λ 2 2 A -3 D P 2 AP 0 2 b e td e 2t t 0 P D AP P p q p q x e ta c e P tdp c P e td P c

12 P c c x 3 2 e 2t te 2t y e 2t c c 2 e 2t 3 3t t 3 c c 2 B λ b 0 c 0 xt C e λt + {bc 2 + a λc {te λt yt C 2 e λt + {cc a λc 2 }te λt x 2y dy 5x + y 3 2 A 5 λ 2 4λ , λ 2 + 3i, λ 2 2 3i Ap λp 3 2 p p 2 + 3i 5 { 3p ip 3ip 2 p + 3i 5,.. + 3i.p 5p i q q 2 3i 5 { 3q 2 2 3iq + 3iq 2 q 3i 5,.. 3i.q 5q + 2 3i 5 P + 3i 3i P, P i i + 3i A D P 2 + 3i 0 AP 0 2 3i x e ta c e P td P c P e td P c 2

13 P c c xt + 3i 3i e λt 0 yt e λ t + 3i c e α+iβt 3i + c c c 2 e α iβt xt 2 cos 3t 6 sin 3te 3t 6 cos 3t 2 sin 3te 3t C + C yt 0 cos 3te 2t 2 0 sin 3te 2t d x a b y c d x y, A a c b d 2.9 A λ α + iβ, λ α iβ p, x c p q 2.6 e λt + c 2 q e λ t c p e α+iβt + c 2 q e α iβt e α+iβt e α iβt e α+iβt + e α iβt 2e αt cos βt e α+iβt e α iβt 2ie αt sin βt 4 p x C 2e αt q cos βt + C 2 2ie αt sin βt 2. P Ap λp { a b p p ap + b α + iβp α + iβ a α iβp b c d cp + d α + iβ... p : b : a α iβ k p bk, {a α iβ}k k a α + iβ p ba α ibβ a α 2 + β 2 q ba α + ibβ a α 2 + β 2 4 3

14 p e α+iβt p e α+iβt e αt ba α ibβ cos βt + i sin βt a α 2 + β 2 q e α iβt { e αt ba α cos βt + bβ sin βt bβ cos βt + ba α sin βt i {a α 2 + β 2 } cos βt {a α 2 + β 2 } sin βt q e α iβt e αt ba α + ibβ cos βt i sin βt a α 2 + β 2 { e αt ba α cos βt + bβ sin βt bβ cos βt + ba α sin βt + i {a α 2 + β 2 } cos βt {a α 2 + β 2 } sin βt } } xt C { ba α cos βt + bβ sin βt}e αt + C 2 {bβ cos βt + ba α sin βt}e αt yt C {a α 2 + β 2 } cos βte αt + C 2 {a α 2 + β 2 } sin βte αt C λ α + iβ, λ α iβ, xt C p e α+iβt + C 2 q e α iβt yt C e α+iβt + C 2 e α iβt 2.2 xt C { ba α cos βt + bβ sin βt}e αt + C 2 {bβ cos βt + ba α sin βt}e αt 2.3 yt C {a α 2 + β 2 } cos βte αt + C 2 {a α 2 + β 2 } sin βte αt p q 6 2 A x 2y dy x + y λ 2 + 0, λ i, i xt 2C cos t sin t + 2C 2 cos t + sin t C cos t + sin t C 2 cos t sin t yt 2C cos t + 2C 2 sin t C cos t + C 2 sin t 4

15 2 y 2y + y 0 y C e x + C 2 xe x 2 y 2e x 3xe x 2 2 y 8y + 5y 0 e 3x, e 5x y C e 3x + C 2 e 5x 2 e 3x + e 5x e 3x e 5x y, y 2,, y n C y + C 2 y 2 + C n y n 0 C C 2 C n 0 y, y 2,, y n 2C C 2 C n 0 y, y 2,, y n y y 2 y n y y 2 y n W y, y 2,, y n... y n y n 2 yn n W 0 y,, y n W 0 3 Ax + bt 3. Ax 2.2 x e ta c 3.,c t ct x e ta ct t AetA ct + e ta dc Ae ta ct + e ta dc dct e ta bt ct e ta bt + C AetA ct + bt C x e ta c + e ta ct e ta c + e ta e ta bt + C e ta e ta bt + c 3.2 c 5

16 7 A 7x 4y + 2et dy 2x 7y + 4et A λ 2 λ λ λ, λ 2 0 D P P D AP 0 p q p q p, 2, q 2, q 3 3 P P 2 P, P e ta P e td P e t 0 0 e t e t + 4e t 6e t + 6e t 2e t 2e t 4e t 3e t e ta e ta 3e t + 4e t 6e t + 6e t 3e t + 4e t 6e t + 6e t 2e t 2e t 4e t 3e t 2e t 2e t 4e t 3e t t t 2e t bt bt 4e t e ta bt 3e t + 4e t 2e t 2e t 6e t + 6e t 4e t 3e t 2e 2t e 2t 4e 2t 2e 2t 2e t 4e t 3.2 xt e ta e ta c bt + yt c 2 3e t + 4e t 2e t 2e t 6e t + 6e t 4e t 3e t e 2t 2e 2t et + c 3e t + 4e t + c 2 2e t 2e t 2e t + c 6e t + 6e t + c 2 4e t 3e t + c c 2 6

17 Bernoulli Riccati Clairaut, Lagrange A y + P xy Qxy n n 0, Qx 0 y ky k / y n y n u } y n e {C W n Qe W, W n P x B y + P xy 2 + Qxy + Rx 0 y {P xy 2 + Qxy + Rx} y y 2 y y y + e W P e W + C, W 2P y + Q C y xp + fp y xy + fy p y y xy + + y 2 y Cx + fc x f C, y Cx + fc Cf C + fc D y xfp + gp y xfy + gy p y p x exp { f p fp p dp C g p } gp p exp f p fp p dp y xfp + gp p 0 fp 0 p 0 y xfp 0 + gp 0 G OOD L U C K! S E E Y OU A G A I N! 7

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さくらの個別指導 ( さくら教育研究所 ) A 2 P Q 3 R S T R S T P Q ( ) ( ) m n m n m n n n 1 1.1 1.1.1 A 2 P Q 3 R S T R S T P 80 50 60 Q 90 40 70 80 50 60 90 40 70 8 5 6 1 1 2 9 4 7 2 1 2 3 1 2 m n m n m n n n n 1.1 8 5 6 9 4 7 2 6 0 8 2 3 2 2 2 1 2 1 1.1 2 4 7 1 1 3 7 5 2 3 5 0 3 4 1 6 9 1

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