December 28, 2018

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1 December 28, 28

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3 Chapter (2..6) f(t, x) x t n n A(t) X : (a, b) R n dx dt = A(t)X (..) X (a, b) (..) A t A(t) t A(t)X R n compact... n = t = x F (t) = t f(s)ds...2. F (t) = dx dt = f(t)x x(t) = e F (t) x, dx = f(t)x + g(t)y dt dy dt = h(t)y t f(s)ds, H(t) = 2 t h(s)ds

4 t x(t) = e F (t) x + (e F (t) ) g(s)e H(s) F (s) ds y y(t) = e H(t) y. K = C R.. ( ). V K- u V u V norm (N) v V u. v = v =. (N2) λ K v V λv = λ v. (N3) u, v V u + v u + v. V 2 c, c 2 > v V c v v 2 c 2 v..2. V K- u, v V d(u, v) = u v d(, ) V V 2 d i (v, u) = v u i d (, ) d 2 (, ) c, c 2 > u, v V c d (u, v) d 2 (u, v) c 2 d (u, v) d(, ) V d(, ) V V..3. V K- V (V, ) Banach space 3

5 ..4. x = (x,..., x n ) R n n x = x i i= x 2 = n x i 2 i= x = max i=,...,n x i, 2, 3 R n x x 2 x n x (R n, p ) Banach R n..5. R n 2. i =,..., n e i R n i M = max i=,...,n e i x = (x,..., x n ) = n i= x ie i x = n x i e i i= n x i e i M x 2 C > x R n x C x 2 i= C 2 > x R n C 2 x 2 x (..) (..) {x m } m R n m x m x m 2 x m y m 2 = m y m = x m x m 2 y m =. x m 2 {y m } m {z k } k z R n m z m z 2. z 2 = z m z z m z C z m z 2 z = lim m z m =. z =. z 2 = (..) 4

6 ..6. (X, d) C b (X, d) = {f f : X R, f X } f C b (X, d) f = sup f(x) x X b C(X, d)..7. V K- A : V V (L) A V. (L2) Av sup < +. v V,v v. (L) (L2): (L2) {x m } m V m Ax m x y m = n x m xm Ax m m x m y m = Ax m, Ay Ax m m = x m A Av (L2) (L): C = sup x V m x m x v V,v v Ax m Ax = A(x m x) C x m x m Ax m Ax. A V..8. V K-..7 (L2) A : V V V V bounded linear operator A L(V ) L(V ) = {A A V V } A L(V ) = Av sup v V,v v L(V ) L(V ) operator norm 5

7 ..9. V K- L(V ) K- L(V ) L(V ) A, B L(V ) AB L(V ) A L(V ) B L(V ) (V, ) Banach (L(V ), L(V ) ) Banach. (N): A L(V ) A L(V ) = v V Av =. A =. (N2): A L(V ), λ K λax = λ Ax λa L(V ) = λ A L(V ). (N3): A, B L(V ) (A + B)v = Av + Bv Av + Bv. A + B L(V ) A L(V ) + B L(V ). L(V ) L(V ) A, B L(V ) Bv ABv = ABv Bv v Bv v AB L(V ) A L(V ) B L(V ). (V, ) Banach {A m } m L(V ) Cauchy ϵ > M ϵ > n, m M ϵ A n A m L(V ) < ϵ v V n, m M ϵ A n v A m v = (A n A m )v A n A m L(V ) v ϵ v. (..2) {A m v} m V Cauchy V Banach m Av A : V V (..2) m n M ϵ A n v Av ϵ v. n M ϵ A n A L(V ) ϵ. n A n A. (L(V ), L(V ) ) Banach... L(R n ) = M n,n (R) = R n2, M m,n (R) m n R n 2 L(R n ) 2 Ax 2 A 2 = sup. 2 L(R n ) = R n2 x R n,x x R n2 2 (L(R n ), 2 ) Banach 6

8 .2 K = R, C V K-.2.. (V, ) Banach {x n } n V N n x n < + x n n n=.2.2. (V, ) {x n } n V n n= x n n x n N x n.2.. y N = N n= x n y N+m y N N+m n=n+ x n. n= x n < + ϵ > N N+m n=n x n n N x n < ϵ. y N+m y N < ϵ {y n } n V Cauchy (V, ) Banack y V n y n y (V, ) Banach A L(V ) A n n! n e A A, B L(V ) AB = BA e A+B = e A e B. A n /n! L(V ) ( A L(V ) ) n /n! a = A L(V ) n= an /n!.2. n An /n! A, B L(V ) AB = BA U N = {(n, m) n, m {,,..., N}, n + m N + } ( N = n= N A n n! k k= r= )( N m= B m m! ) = N A k r B r (k r)!r! + N n= m= (n,m) U N An Bm n!m! A n B m n!m! = N (A + B) k + k! k= (n,m) U N n= An Bm n!m!. (.2.) 7

9 α = A L(V ), β = B L(V ) L(V ) (n,m) U N An Bm n!m! (n,m) U N αn βm n!m! k N+ (α + β) k k! N = e α+β (α + β) k k! k= N (n,m) U N e A e B = e A+B. A n B m n!m! (.2.) N.2.4. (V, ) a, b R a < b f : (a, b) V f α (a, b) v V lim f(α + h) f(α) h v h = v = df (α) f α dt.2.5. (V, ) Banach A L(V ) t R U(t) = e ta t R U(t) du = AU(t) = U(t)A dt. U(t + h) U(t) h B(h) = n 2 = e(t+h)a e ta h n 2 An h n! = e ta eha I h ( = e ta A + h h n 2 = eha I ( e ta = A + h h h n 2 h n 2 An n 2 An ) n! ) e ta n! (.2.2) B(h) L(V ) n 2 h n 2 ( A L(V )) n n! e A L(V ). (.2.2) h du dt = AU(t) = U(t)A. 8

10 .3 A M n,n (K) dx = Ax (.3.) dt K = C or R.3.. (.3.) t = x K n x(t) = e ta x e ta Jordan.3.2. k N I k, J k M k,k (C) I k J k = (Jordan ). A M n,n (C) P M n,n (C) n, n 2,..., n m N, A α,..., α m, P n n m = n α I n + J n.... P α AP = 2 I n2 + J n (.3.2)..... α m I nm + J nm C n (φ,..., φ n, φ 2,..., φ 2 n 2,..., φ m,..., φ m n m ) k =,..., m { Aφ k λφ k i i =, i = λφ k i + φ k i i = 2,..., n k 9

11 P AP A Jordan α i I ni + J ni A Jordan Jordan cell i =,..., m n i = m = n A diagonlizable semisimple.3.5. P, A M n,n (C) P e P AP = P e A P t t 2 t... k 2! (k )!. t... e tj k = t 2 2!... t... e ta = P e ta P e αt e tj n... e ta e = P α2t. e tjn P (.3.3)... e αmt e tj nm k =,..., m, i =,..., n k B = P AP i e ta φ k i = e α t kt j j! φk i j j= K n P e ta K n P (.3.4) e tb K n K n

12 .3.8. x(t) (.3.) (, ) x() x(t) = x(t) = e ta x() x() = m n k k= m n k a k i φ k i k= i= ( nk i e α kt a k t j i+j j! i= j= A k =,..., m n k = ψ k = φ k m x() = a k ψ k x(t) = k= m a k e αkt ψ k k= A M n,n (R) λ / R λ = η + ω.3.9. E(λ) = {x x C n, m (A λi) m x = } φ E(λ), e φ = f + g, f, g R n f g. φ = f g φ E(λ). E(λ) E(λ) φ φ f = (φ + φ)/2, g = (φ φ)/(2 ) f g.3.. (φ,..., φ m ) λ Jordan Aφ = λφ, Aφ i = λφ i + φ i (i = 2,..., n) φ i = f i + gi, f i, g i R n k e ta f k = e ηt t j j! (cos ωtf k j sin ωtg k j ) j= k e ta g k = e ηt t j j! (sin ωtf k j + cos ωtg k j ) j= ) φ k i

13 . Lemma.3.6 e ta (f k + k g k ) = e λt t j j! (f k j + g k j ).3.. j= P = P = a, b R a B = a b e at te at e tb = e at e bt A = P BP a + 2b + 2 a b 2a + 2b + 3 A = 2a 2b + 4 b 2 2a 2b + 6 2a 2b a + b 3a 2b ( + 2t)e at + 2e bt ( t)e at e bt ( 2 + 3t)e at + 2e bt e ta = (2 + 4t)e at 2e bt 2te at + e bt (2 + 6t)e at 2e bt 2e at 2e bt e at + e bt 3e at 2e bt a = b = 3 + 2t t 3t A = 4 3 6, e ta = e t 4t 2t 6t 2

14 a =, b = A = P.3. α, η, ω R η ω B = ω η α B η ± ω, α z = η + ω B z C C { t(f + } g) f =, g =, t C z { t(t } g) t C..3. e tb f = e ηt ((cos ωt)f (sin ωt)g) e tb g = e ηt ((sin ωt)f + (cos ωt)g) e ηt cos ωt e ηt sin ωt e tb = e sin ωt e ηt cos ωt e αt A = P BP η 2ω + 2α η + ω α 2η 3ω + 2α A = 2η 6ω 2α 4ω + α 2η ω 2α 2η ω 2α η + ω + α 3η 2ω 2α 3

15 e ta = P e tb P e ta A η, ω, α e ηt cos ωt, e ηt sin ωt, e αt ω = η = α = 3 A = 6 5, cos t + 2 sin t + 2 cos t sin t 2 cos t + 3 sin t + 2 e ta = e t 2 cos t + 6 sin t 2 4 sin t + 2 cos t + sin t 2 2 cos t + sin t cos t sin t + 3 cos t + 2 sin t 2.4 A 3 dx dt = Ax x() A A λi = λ, λ 2, λ 3 (a) λ, λ 2, λ 3 (a-) λ, λ 2, λ 3 p, p 2, p 3 λ, λ 2, λ 3 x(t) x(t) = αe λ t p + βe λ 2t p 2 + γe λ 3t p 3 P = (p p 2 p 3 ) α β = P x(). γ (a-2) λ, λ 2 λ = η + ω (η, ω R) λ 2 = η ω λ 3 λ u + v (u, v R 3 ) u v λ 2 A(u + v) = λ (u + v) A(u v) = λ 2 (u v) 4

16 .3. e ta u = e ηt ((cos ωt)u (sin ωt)v) e ta v = e ηt ((sin ωt)u + (cos ωt)v) λ 3 p e ta p = e λ 3t p x() = αu + βv + γp x(t) = e ηt (α cos ωt + β sin ωt)u + e ηt (β cos ωt α sin ωt)v + e λ 3t p Q = (u v p) α β = Q x(). γ (b) λ = λ 2 λ = λ 2 λ 3 (b-) λ p, p 2 λ p 3 λ 3 x(t) = αe λ t p + βe λ t p 2 + γe λ 3t p 3. t = x() = αp + βp 2 + γp 3 P = (p p 2 p 3 ) α β = P x(). γ (b-2) λ (A λ I)p (A λ I) 2 p = p p 2 = p p = (A λ )p 2 (A λ I)p = (A λ I) 2 p 2 = 5

17 Ap = λ p. p 2 λ (p, p 2 ) p 3 λ 3 Ap = λ p, Ap 2 = λ p 2 + p, Ap 3 = λ 3 p 3 P = (p p 2 p 3 ) λ P AP = λ λ 3 e λ t te λ t e ta = P e λ t P e λ 3t e ta p = e λ t p, e ta p 2 = e λ t p 2 + te λ t p, e ta p 3 = e λ 3t p 3 x() = αp + βp 2 + γp 3 e ta x() = e λ t (α + βt)p + βe λ t p 2 + γe λ 3t p 3. α β = P x(). γ (c) λ = λ 2 = λ 3 λ = λ = λ 2 = λ 3 λ E(λ) = {x x R 3, (A λi)x = } (c-): dim E(λ) = 3 A λi = λ Jordan λ λ (c-2): dim E(λ) = 2 A λi, (A λi) 2 = λ Jordan λ λ (c-3): dim E(λ) = A λi, (A λi) 2, (A λi) 3 = 6

18 λ Jordan λ λ (c-) A = λi x(t) = e λt x(). (c-2) (A λi)p p p 2 = p p = (A λi)p 2 Ap = λp. dim E(λ) = 2 p 3 E(λ) (p, p 3 ) P = (p p 2 p 3 ) λ P AP = λ λ t e ta = e λt P P (b-2) λ = λ 3 (c-3) (A λi) 2 p p p 3 = p p 2 = (A λi)p 3, p = (A λi)p 2 (A λi)p = (A λi) 3 p 3 = Ap = λp, Ap 2 = λp 2 + p, Ap 3 = λp 3 + p 2 P = (p p 2 p 3 ) λ P AP = λ λ e ta = e λt P t t2 2 t P 7

19 e ta p = e λt p, e ta p 2 = e λt (tp + p 2 ), e ta p 3 = e λt ( t 2 x() = αp + βp 2 + γp 3 x(t) = e ta x() = e λt ( ( α + βt + γ t2 2 2 p + tp 2 + p 3 ) ) p + (β + γt)p 2 + γp 3 )..4.. () 3 3 A = 2 2 A λi = (λ ) 3. (c) 2 3 A I = s 2. (c-3) 2 2 (A I) 2 = (A I) 3 = p 3 =, p 2 = (A I)p 3 =, p = (A I) 2 p 3 = 2 8

20 (2) P = 2 P = t t2 t e ta = e t + t2 2 3t t 2 2t 2 + t 2 2t 2t 2 2t t 2t A = A λi = ( λ)(λ 2 + 2λ + 2) λ = + i, λ 2 = i, λ 3 =. (a-2) λ 2 2i i λ 3 = u = 2, v = 2, p 3 = P = (u v p 3 ) = P =

21 .5 K = R or C (.3.) dx dt = Ax x(t) = A Jordan n,..., n m (n n m = n), α,..., α m η,..., η m A Jordan B A Jordan P B = P AP.5.. β R i =,..., m η i < β C > t e ta Ce bt L(R n ). e ta (i, j) a ij (t) a ij (t) = m e ηkt e n k ω k t c kl t l k= β > η k C t t k e ηt Ce βt C ij > a ij (t) C ij e βt M n,n (R n ) R n2 C > Z M n,n (R n ) l= Z L(R n ) C Z e ta C e ta = C max a ij (t) ( C max C ij ) e βt 2

22 .5.2. () (.3.) x (stable) t e ta as t (2) (.3.) x C > t e ta C (3) (.3.) x x() t e ta x().5.3. () (.3.) x i =,..., m η i < (2) (.3.) x i =,..., m η i < η i = n i = (3) (.3.) x i =,..., m η i > η i = n i 2. () i =,..., m η i < β < i =,..., m η i < β <.5. t e ta. i η i α i u i e ta u i = e α it u i e ta u i = e η it u i u i (2), (3) (a) i =,..., m η i < η i = n i = i =,..., k η i <, η k+ =... = η m = e ta (p, q)- a pq (t) a pq (t) = k i= n i e α it j= c p,q ij tj + m i=k+ c p,q i e α it i =,..., k η i < β < β C pq > m a pq (t) C pq e βt + c pq i i=k+.5. C > t e ta C. (b) i =,..., m η i > η i = n i 2 η i > α i u i e ta u i = e α it t e ta u i = e η it u i 2

23 η i = n i φ, φ 2 e ta φ = e α it (φ + tφ 2 ) t e ta = φ + tφ 2..6 K = R or C f : R K n A M n,n (K) dx dt = Ax + f(t) (.6.5) x(t) f (t). f(t) =. i =,..., n f i f n (t) [a, b] f i (.6.5) f(t) (external force) (non homogeneous term).6.. (.6.5) x(). (.6.5) x(t) = e ta x() + t d dt (e ta x(t)) = e ta f(t) e (t s)a f(s)ds (.6.6) 22

24 t e (t s)a f(s)ds (.6.5) (.6.5) x() = ( dx ) ( ) dt = Ax x() + (.6.5) c,..., c n d n dt φ + c d n n n dt φ c d n dt φ + c φ = (.6.7) φ : R K (.6.7) n homogeneous constant coefficient ordinary differential equation of order n) x = φ, x 2 = d dt φ,..., x n = dn dt n φ x x =..... A = c c c n 2 c n x n (.6.8) (.6.7) d dt x = Ax (.6.7) x() = φ() φ () (). φ(i) = φ (n ) () 23

25 d i dt i φ x(t) = e ta x() e ta (i, j)- a ij (t) n φ(t) = φ (j ) ()a j (t).6.2. () A p A (λ) = λi A j= p A (λ) = λ n + c n λ n + + c (2) A λ λ E A (λ) (3) λ,..., λ m (i j λ i λ j ) A p A (λ) = (λ λ ) n (λ λ m ) nm A Jordan λ I n + J n.... λ 2 I n2 + J n (4) k... λ m I nm + J nm ( d dt λ ) k+(t k e λt ) = (4), i =,..., m, j =,..., n i φ ij (t) = tj (j )! eλ it (.6.7) φ m n i φ(t) = c ij φ ij (t) (.6.9) i= (c ij ) i=,...,m,j=,...,ni (.6.9) φ (.6.7) j= 24

26 .6.3. () (φ,..., φ n ) (.6.7) (a fundamental system of solutions) i =,..., n φ i (.6.7) (.6.7) φ α,..., α n φ(t) = n α i φ i (t) i= (2) i =,..., n u i : R K n U = (u,..., u n ) U (Wronski matrix) M W U (t) u (t) u 2 (t) u n (t) MU W u () (t) u () 2 (t) u () n (t) (t) =... u (n ) (t) u (n ) 2 (t) u (n ) n (t) U (Wronski determinant or Wronskian) W U (t) W U (t) = M W U (t) (4) (φ,..., φ n,..., φ m,..., φ mnm ) (.6.7).6.5. Φ = (φ,..., φ n ) (.6.7) t R MΦ W (t) φ(t) = n α i φ i (t) i= α φ(t) MΦ W α 2 (t). = φ () (t). φ (n ) (t) α n 25

27 φ() φ () () (.6.7). φ (n ) () α φ() α 2. = M Φ W () φ () (). φ (n ) () a.. a n α n φ(t) = n α i φ i (t) i= K n b b n b. = e ta. b n a a n K n. (.6.7) ψ(t) ψ(t) a. =. ψ (n ) (t) a n ψ (.6.7). ψ(t) = i= β β n β i φ i (t) 26

28 β MΦ W (t). β n = ψ(t) a. =. ψ (n) (t) M W Φ (t) : Kn K n M W Φ a n. (t).6.6. R, C, L R, L >, C > I d 2 dt I + R d 2 L dt I + LC I = (.6.) J = d dt I d dt ( ) ( I = J LC R L ) ( ) I J λ 2 + R L λ + =. (.6.) LC ( R ) 2 4 L LC. () > CR 2 > 4L (.6.) ( λ = ) ( R (R ) 2 2 L + 4 and λ 2 = ) R (R ) 2 L LC 2 L 4 L LC λ, λ 2 < Φ = (e λt, e λ2t ) (.6.) ( ) e MΦ W λ t e (t) = λ 2t λ e λ t λ 2 e λ 2t W Φ (t) = e (λ +λ 2 )t (λ 2 λ ) (I(), I ()) I(t) = α e λt + α 2 e λ2t ( ) ( ) ( ) α I() = λ λ 2 α 2 I () 27

29 λ, λ 2 < (.6.) (2) = CR 2 = 4L (.6.) λ = R 2 L. p A (λ) = (λ λ ) 2 Φ = (e λt, te λt ) (.6.) ( ) e MΦ W λ t te (t) = λ t λ e λ t e λt + λ te λ t W Φ (t) = e 2λ t (I(), I ()) I I(t) = α e λ t + α 2 te λ t ( ) ( ) α = λ α 2 ( ) I() I () λ < (.6.) (3) < CR 2 < 4L η = R 2 L and ω = LC R ) 2 4( L (.6.) η ± ω C (e (η+ ω)t, e (η ω)t ) e (η+ ω)t = e ηt (cos ωt + sin ωt) e (η ω)t = e ηt (cos ωt sin ωt) Φ = (e ηt cos ωt, e ηt sin ωt) ( ) MΦ W e (t) = ηt cos ωt e ηt sin ωt ηe ηt cos ωt ωe ηt sin ωt ηe ηt sin ωt + ωe ηt cos ωt W Φ (t) = e 2ηt ω 28

30 (I(), I ()) I(t) = e ηt (α cos ωt+α 2 sin ωt) ( ) ( ) ( ) α I() = η ω I () α 2 R > η < (.6.) R = (.6.) f : R K d n dt φ + c d n n n dt φ c d n dt φ + c φ = f(t) (.6.2) φ (.6.2) f n A (.6.8) F (t) =. f(t).6. d x = Ax + F (t) dt x(t) = e ta x() + t e (t s)a F (s)ds e ta (i, j)- a ij (t) t t a n (t s) e (t s)a F (s)ds = f(s). ds a nn (t s) a n (t). a n n (t) = eta. a nn (t) 29

31 a n (t) (.6.7) (.6.7). ϕ(t) φ (t) = t ϕ(t s)f(s)ds φ (.6.2). (.6.2). γ φ(t) (.6.7). γ n ψ(t) φ(t) = ψ(t) + φ (t) φ (t) Φ = (φ,..., φ n ) (.6.7) () α. α n = W Φ(). ϕ(t) = α n n α i φ i (t) φ (t) = i= t ϕ(t s)f(s)ds. (2) : φ (t) = n i= ϕ i(t)φ i (t) ϕ i (t) γ γ n 3

32 .6.8. WΦ,f(t) k = φ (t)... φ k (t) φ k+ (t)... φ n (t) φ () (t)... φ () k (t) φ() k+ (t)... φ() n (t)..... (t)... φ (n 2) k (t) φ(n 2) k+ (t)... φ(n 2) n (t) (t)... φ (n ) (t) f(t) φ(n ) (t)... φ(n ) n (t) φ (n 2) φ (n ) k i =,..., n ϕ i (t) = φ (t) = t W i Φ,f (t) W Φ (t) dt n ϕ i (t)φ i (t). i= k+. (ϕ,..., ϕ n ) ϕ (t) MΦ W (t). ϕ n (t) =. ϕ n(t) f(t) n i= ϕ i(t)φ (j) i (t) = n φ (j) i= (t) = n = φ (n) i= { (j =,..., n 2) f(t) j = n ϕ i (t)φ (j) i (t) (j =,..., n ) ϕ i (t)φ (n) i (t) + f(t) j = n (.6.3) (.6.4) (t) + c n φ (n ) (t) + + c φ () (t) + c φ (t) n ϕ i (t) ( φ (n) i (t) + c n φ (n ) i (t) + + c φ () i (t) + c φ i (t) ) + f(t) = f(t) i= 3

33 φ (t) (.6.2) (.6.4) (.6.) f(t) d 2 dt I + R d 2 L dt I + LC I = f(t). d 2 dt I + R d 2 L dt I + I = cos γt (.6.5) LC.6.6 (.6.) () > CR 2 > 4L (.6.) λ = 2 ( R L + (R L ) 2 4 LC (.6.) ) and λ 2 = 2 α 2 e λ2t ( ) ( ) α = λ λ 2 α = t λ 2 λ, α 2 = ϕ(t) = e λs cos γsds = φ (t) = ( R L (R L ) ) 2 4. LC ( ) ϕ ϕ(t) = α e λt + α 2 λ 2 λ. ( ) λ 2 λ ( e λ t + e λ 2t ) t ϕ(t s) cos γsds e λt λ ( λ cos γt + γ sin γt) + λ 2 + γ2 λ 2 + γ 2 32

34 φ (t) = (λ λ 2 γ 2 ) cos γt γ(λ + λ 2 ) sin γt (λ 2 + γ 2 )(λ γ 2 ) ( λ + λ 2 λ λ 2 + γ 2 eλ t + λ ) 2 λ γ 2 eλ 2t A = (λ 2 + γ 2 )(λ γ 2 ) θ cos θ = λ λ 2 γ 2, sin θ = γ(λ + λ 2 ) A A φ (t) = A cos (γt + θ) + ( λ λ 2 λ λ 2 + γ 2 eλ t + λ ) 2 λ γ 2 eλ 2t I(t) λ, λ 2 < t I(t) = c e λ t + c 2 e λ 2t + φ (t) I(t) cos (γt + θ) A (2) = CR 2 = 4L (.6.) λ = R 2 L. ( ) (.6.) ϕ(t) = α e λt + α 2 te λt ( ) ( ) ( ) α = λ α 2 α =, α 2 =. ϕ(t) = te λ t. φ (t) = t = te λ t = ϕ(t s)f(s)ds t e λ s cos γsds e λ t t se λ s cos γsds ( ) (λ 2 (λ 2 + γ 2 ) 2 γ 2 ) cos γt 2λγ sin γt 33 + γ2 λ 2 + λ t λ 2 + γ 2 e λ t.

35 A = λ 2 + γ 2 cos θ = λ2 γ2, sin θ = 2λ A γa θ φ (t) = A cos (γt + θ) + γ2 λ 2 + λ t λ 2 + γ 2 e λ t. I(t) I(t) = c e λ t + c 2 te λ t + φ (t) λ < t I(t) cos (γt + θ). A (3) < CR 2 < 4L η = R 2 L and ω = LC R ) 2 4( L (.6.) ( ) (e ηt cos ωt, e ηt sin ωt) (.6.) ϕ(t) ϕ(t) = α e ηt cos ωt+α 2 e ηt sin ωt ( ) ( ) ( ) α = η ω α 2 α =, α 2 = eηt. ϕ(t) = ω ω sin ωt. φ (a) η ω γ Q = (η 2 + (ω γ) 2 )(η 2 + (ω + γ) 2 ) = (η 2 ω 2 + γ 2 ) 2 + 4η 2 ω 2 ( φ (t) = eηt (η 2 + ω 2 γ 2 ) cos ωt + η ) Q ω (η2 + ω 2 + γ 2 ) sin ωt η = R = φ (t) = (b) η = ω = γ + (η2 + ω 2 γ 2 ) cos γt 2ηγ sin γt Q (cos γt cos ωt) ω 2 γ2 φ (t) = t sin ωt 2ω 34

36 f d 2 dt f + a d 2 dt f + a f = (.6.6) φ (c a )(a c a c ) + (c a ) 2 (.6.7) d 2 dt φ + c d 2 dt φ + c φ = f(t) α f + α 2 f (.6.6) f = a f a f φ = α f + α 2 f d 2 dt φ + c d 2 dt φ + c φ f = (a 2 a )f + a a f = α ((a 2 a ) a c +c )f +(a a c a )f ( ) )+α 2 (c a )f +(c a )f ( ) ( ) a (c a ) + c a c a α = a (c a ) c a α 2 ( ) (.6.8) φ (.6.6) (.6.8) (.6.7) 35

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