y + P (x)y + Q(x)y = R(x) (1) R I. P (x), Q(x) R(x) I. 2 (p(x)y ) + q(x)y = f(x) (2)., c I, p(x) = e R x c P (t)dt > 0, (1) (2). (2) (1). L

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Download "y + P (x)y + Q(x)y = R(x) (1) R I. P (x), Q(x) R(x) I. 2 (p(x)y ) + q(x)y = f(x) (2)., c I, p(x) = e R x c P (t)dt > 0, (1) (2). (2) (1). L"

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1 y + P ()y + Q()y = R() (1) R I P (), Q() R() I 2 (p()y ) + q()y = f() (2), c I, p() = e R c P (t)dt >, (1) (2) (2) (1) Lioville, (53 ) (2) (1) R() (2) f(),, 52 (p()y ) + q()y = (3) I = [, b] Ly = (p()y ) + q()y 51 u, v C 2 (I), C L(y 1 + y 2 ) = Ly 1 + Ly 2 L(Cy 1 ) = CLy 1 56

2 51 51 (3) u(), v() (3), 51, C 1, C 2 C 1 u 1 () + C 2 u 2 () (3) (1) u(), v() C 1 u() + C 2 v(), C 1 = C 2 =, (C 1, C 2 ) \= (, ) C 1, C 2 C 1 u() + C 2 v(), u(), v() (2) (3) 2 u(), v(), u(), v() ( ) 2 u(), v() W (u, v)() = u() u () v() v () u(), v() 52 (1) W (u, v)( ) \= u(), v() (2) u(), v() W (u, v) = (1) C 1, C 2 C 1 u()+c 2 v(), C 1 u ()+C 2 v () 2 { C1 u() + C 2 v() = C 2 u () + C 2 v () =, = ( u( ) v( ) u ( ) v ( ) )( C1 C 2 ) = ( ) ( ) 1 u( ) v(, ) u ( ) v, ( ) t (C 1, C 2 ) = t (, ) 57

3 (2), C u() = Cv() v() = Cu() 52(2) ( 53 ), u(), v() (3) 53 u(), v() (3), C > W (u, v)() = C p() u(), v() (3) : (1) u(), v() W (u, v) (2) W (u, v), u(), v() 53 u(), v() W (u, v), u() = { { 2 <, v() = 2 < 54 y + y + by = 2 u(), v() C > W (u, v) = Ce (Hint: W W + W = ), (3) C 1 u() + C 2 v() 55 I = [, b] 1 Lu =, u( ) = A, u ( ) = B, 58

4 56 (3) u(), u u( ) =, u ( ) =, 57 u 1, u 2 C 2 (I) (3) (3) u, C 1, C 2 u() = C 1 u 1 () + C 2 u 2 (), 2,, 2, 1 u(), 1 v(), u(), v() 51 u = c (3), v v() = u() ds c p(s)u 2 (s), v (3), u(), v(), v Cu() (3), C U v() = U()u() v () = U ()u() + U()u () p()v () = p()u ()u() + U()(p()u ()) (p()v ()) = (p()u ()) u() + 2p()U u + U()(p()u ()), u() (3) (p()v ()) + q()v() = (p()u ()) u() + 2p()U ()u () + U()(p()u ()) + q()u()u() = (p()u ()) u() + 2p()U ()u () + U(){(p()u ()) + q()u()} (p()u ()) u() + 2p()U ()u () Lv =, (pu ) u+2pu u = u, (pu ) u 2 +2(pU )uu = 59

5 d d ((pu )u 2 ) =, (pu )u 2 = C C = 1 U = 1 pu 2 y + P ()y + Q()y = v() = u() c e R s c P (t)dt u(s) 2 ds, U = ds c p(s)u 2 (s) 53 : y + y + by = (4),, 58 λ e λ λ = µ + iν (µ, ν ), e λ = e µ (cos ν + i sin ν) e λ = e µ > e λ (4) y = e λ y = λe λ, y = λ 2 e λ (λ 2 + λ + b)e λ = 58 λ 2 + λ + b = (5) 2 (4) 2 4b, 3 (1) 2 4b >, (5) 2 λ = λ 1, λ 2 e λ1 e λ2,, 6

6 (2) 2 4b =, (5) 2 λ = λ = 2, eλ (4), 51, e λ e R s dt ds = (e λs e 2 ) 2 e s (e (/2)s ) 2 ds = eλ ds = e 2 = e λ, e λ e λ, (3) 2 4b <, (5) 2 λ 1 = µ + iν,λ 2 = µ iν, e λ1 = e µ (cos ν + i sin ν), e λ 2 = e µ (cos ν i sin ν), e λ 1 + e λ 2 2 = e µ cos ν, e λ 1 e λ 2 2i = e µ sin ν e µ cos ν, e µ sin ν 59 2 y + y + by = λ 2 + λ + b = (1) λ = λ 1, λ 2, e λ1, e λ 2 (2) 2 λ = λ, e λ, e λ (3) λ = µ ± iν, e µ cos ν, e µ sin ν 54 y + P ()y + Q()y = f() (6) 61

7 (p()y ) + q()y = f() (7), y + P ()y + Q()y = (8) (p()y ) + q()y = (9) u 1, u 2 y = C 1 u 1 () + C 2 u 2 () (C 1, C 2 ), : 51 (7) 1 ( ) w, (7), y = w + C 1 u 1 + C 2 u 2 = + (7) v Lu = (p()u ) + q()u, L(v w) = Lv Lw = f() f() = v w v w = C 1 u 1 + C 2 u 2 v = w + C 1 u 1 + C 2 u 2 51,, 1,, C 1 u 1 () + C 2 u 2 () (C 1, C 2 ) C 1, C 2 U 1, U 2, (9) u 1, u 2, (9) y = C 1 u 1 + C 2 u 2 (C 1, C 2 ) C 1, C 2 U 1, U 2 y = U 1 u 1 + U 2 u 2 (7) U 1, U 2, y = U 1u 1 + U 1 u 1 + U 2u 2 + U 2 u 2 = (U 1u 1 + U 2u 2 ) + (U 1 u 1 + U 2 u 2) 62

8 , U 1, U 2 : U 1u 1 + U 2u 2 = (1) y = U 1 u 1 + U 2 u 2, p()y = U 1 (p()y ) + U 2 (p()y ) (p()y ) =U 1(p()u 1) + U 1 (p()u 1) + U 2(p()u 2) + U 2 (p()u 2) = {U 1(p()u 1) + U 2(p()u 2)} + U 1 (p()u 1) + U 2 (p()u 2) (11), (11) (p()y ) + q()y = {U 1(p()u 1) + U 2(p()u 2)} + U 1 (p()u 1) + U 2 (p()u 2) + q()(u 1 u 1 + U 2 u 2 ) = {U 1(p()u 1) + U 2(p()u 2)} + U 1 {(p()u 1) + q()u 1 } + U 2 {(p()u 2) + q()u 2 }, u 1, u 2 (9) (p()u 1 ) +q()u 1 =, (p()u 2) +q()u 2 = (p()y ) + q()y = {U 1(p()u 1) + U 2(p()u 2)}, y (7) f(), {U 1(p()u 1) + U 2(p()u 2)} = f() (12) (1), (12) U 1, U 2, ( )( ) ( ) u1 u 2 U 1 p()u 1 p()u = 2 f() U 2 p()w (u 1, u 2 ) p() >, u 1, u 2 54, p()w (u 1, u 2 ), ( ) ( )( ) U 1 = 1 p()u 2 u 2 p()w (u 1, u 2 ) p()u 1 u 1 f() U 2 = 1 p()w (u 1, u 2 ) ( ) u2 f() u 1 f(), U 1() = u 2()f() p()w (u 1, u 2 ), U 2() = u 1 f() p()w (u 1, u 2 ) 63

9 , U 1 () = U 1 (c) + c c u 2 (s)f(s) p(s)w (u 1, u 2 ) ds, U 2() = U 2 (c) = U 1 (c), U 2 (c) C 1, C 2 ( ) ( u y = C (s)f(s) p(s)w (u 1, u 2 ) ds u 1 () + C 2 + c =u 1 () c u 2 (s)f(s) p(s)w (u 1, u 2 ) ds + u 2() c c u 1 (s)f(s) p(s)w (u 1, u 2 ) ds ) u 1 (s)f(s) p(s)w (u 1, u 2 ) ds u 2 () u 1 (s)f(s) p(s)w (u 1, u 2 ) ds + C 1u 1 () + C 2 u 2 () 53, K = p()w (u 1, u 2 ) 52 y + y = 1 cos y + y = λ = λ = ±i, C 1 cos + C 2 sin cos sin W (cos, sin ) = sin cos 2 cos, sin = 1, sin 1 cos 1 u() = cos cos d + C sin cos d + C 1 2 = cos sin cos d + C 1 cos + sin d + C 2 sin = cos log cos + C 1 cos + sin + C 2 sin = cos log cos + sin + C 1 cos + C 2 sin (C 1, C 2 ) 55 Green, Green, f, 64

10 2 =, = b, f,,, b u() f u = f() u() ( ),, 1 f, y =, = b, =, = b, u() = u(b) = 53 { u () = f() u() = u(b) = (13) u = u() = u 1, u(b) = u 2 ( u 1 u 2!), u 1 () =, u 2 () = b,, W (u 1, u 2 ) = u 1 u 2 u 1 u = b = b u = C 1 u 1 + C 2 u 2,, C 1, C 2, u = U 1 u 1 + U 2 u 2 u = f U 1() = u 2()f() K = (b )f() b, U 2() = u 1()f() K 65 = ( )f() b

11 , U 1(ξ) [, b], U 2(ξ) [, ] U 1 (b) U 1 () = (b ξ)f(ξ) b dξ, U 2 () U 2 () = (ξ )f(ξ) K dξ U 1 () = U 1 (b) + (b ξ)f(ξ) b dξ, U 2 () = U 2 () + (ξ )f(ξ) b dξ u() = ( ) { U 1 (b) + (b ξ)f(ξ) b } { dξ + (b ) U 2 () + (ξ )f(ξ) b } dξ u() = u(b) = U 1 (b) =, U 2 () =, u() = ( ) (b ξ)f(ξ) b dξ + (b ) (ξ )f(ξ) b dξ (14), ( )(b ξ) ( ξ) (b ) G(, ξ) = (b )(ξ ) (ξ b) (b ) u() = G(, ξ)f(ξ)dξ G (13) Green Grenn 2 Lu = (p()u ) + q()u (p()u ) + q()u = f() 1 u() + 2 (p()u ()) = b 2 u(b) + b 2 (p(b)u (b)) = (15), ( 1, 2 ), (b 1, b 2 ) , b b 2 2, α, β, { cos α u() sin α p()u () = cos β u(b) sin β p(b)u (b) = (16) 66

12 (16) Green, (p()u ) + q()u = ( < < b), u 1 () = sin α, p()u () = cos α u 1 (), u 2 (b) = sin β, p(b)u 2(b) = cos β u 2 () ( ), C 1, C 2 u() = C 1 u 1 () + C 2 u 2 (), C 1, C 2, (p()u ) + q()u = f() y = C 1 u 1 () + C 2 u 2 () + 1 K { u 1 () f(ξ)u 2 (ξ)dξ + } f(ξ)u 1 (ξ)dξ, K = p(){u 1()u 2 () u 1 ()u 2()}, I = [, b] ( 53 ) 55 U() = 1 { } u K 1 () f(ξ)u 2 (ξ)dξ + f(ξ)u 1 (ξ)dξ (p()u ) + q()u = f() u { u() = u 1 () C K } { f(ξ)u 2 (ξ) + u 2 () C K } f(ξ)u 1 (ξ)dξ 67

13 u () = u 1() { C K } { f(ξ)u 2 (ξ)dξ + u 2() C K } f(ξ)u 1 (ξ)dξ,, C 1, C 2, cos α u() sin α p()u () = cos α u() sin α u () = {cos α u 2 () sin α p()u 2()} { C K } f(ξ)u 1 (ξ)dξ u 1 u 2, cos α u 2 () sin α p()u 2() ( 56 ) C 2 = 1 K f(ξ)u 1 (ξ)dξ = b C 1 = 1 K f(ξ)u 2 (ξ)dξ u() = 1 { u() K f(ξ)u 2 (ξ)dξ + u 2 () } f(ξ)u 1 (ξ)dξ, u 1 G(, ξ) = K u 1()u 2 (ξ) ( ξ) 1 K u (17) 1(ξ)u 2 () (ξ < b) u() = G(, ξ)f(ξ)dξ G(, ξ) (15) Green 56 u 1, u 2, u 2 cos α u 2 () sin α p()u 2() = 68

14 , u 1 u 2, C 1, C 2, C 1 u 1 + C 2 u 2 =, A, B (p()u ) + q()u = u() = A, u () = B h >, [, + h) 1, A = B =, u (1) U() = C 1 u 1 () + C 2 u 2 (), ( ) ( U() u1 () u p()u = 2 () () p()u 1() p()u 2() )( C1 C 2 ) (2) u 1 () = sin α, p()u 1() = cos α, C 1, C 2, U U() =, p()u () = Hint: A, A = A = (3), (2) C 1, C 2 C 1 u 1 () + C 2 u 2 (), u 1, u 2, Green 69

15 51 (Green ) Green G(, ξ) (1) G(, ξ) (, ξ), G(, ξ) = G(ξ, ) (2) ξ [, b], G(, ξ), = ξ, (p()g (, ξ)) + q()g(, ξ) = f() (3) ξ (, b), G(, ξ), : cos α G(, ξ) p() sin α G (, ξ) = cos β G(b, ξ) p(b) sin β G (b, ξ) = (4) G \= ξ, = ξ (jump) G (ξ +, ξ) G (ξ, ξ) = 1 p(ξ) 511 (15) Green (17) G(, ξ) u() = G(, ξ)f(ξ)dξ f, (Gf)() = G G(, ξ)f(ξ)dξ Green 2 G(, ξ) 1, 2 G(, ξ) Green Green, u 1 u 2, 7

16 , Green ( ( ) ) 54 { u + u = f() u () = u(π) = Green, f() = cos, α = π 2, β = u 1, u + u =, u() = 1, u () = u + u =, u(π) =, u (π) = 1 u 2 u + u = u() = C 1 cos + C 2 sin u 1 () = cos, u 2 () = sin, K = u 1()u 2 () u 1 ()u 2() = 1, { cos sin ξ ( ξ) G(, ξ) = cos ξ sin (ξ < π), f() = cos, u() = G(, ξ)f(ξ)dξ = (cos ξ sin ) cos ξdξ + = sin cos 2 ξdξ + cos sin ξ cos ξdξ = sin [ ] [ ] π ξ sin 2ξ cos 2ξ = sin + + cos = 2 sin 1 cos + sin 2 sin + cos 2 cos 4 4 = 2 sin 1 cos(2 ) cos + = sin (cos sin ξ) cos ξdξ 1 + cos 2ξ 2 dξ + cos sin 2ξ 2 dξ 58 { u + u = f() u() = u (π) = 71

17 Green, f() = sin, 59 Green { u u = f() u() = u(1) =, { u + u = f() u() = u(π) = (18), Green, α =, β = u 1, u + u =, u() =, u () = 1 u + u =, u(π) =, u (π) = 1 u 2, u + u = u() = C 1 cos + C 2 sin,, u 1 () = sin, u 2 () = sin, u 1 u 2,, f f() sin d \=,,, u, f() sin d = {u () + u()} sin d = = [u () sin ] π u () cos d + = u () cos d + u () sin + u() sin d = [ u() cos ] π u() sin d + u() sin d u() sin u() sin d = f, u() = C sin (C ),, (18) f() sin d = 72

18 ,, f sin, 56,, f, 512 (p()u ) + q()u = f() cos α u() sin α p()u () = cos β u(b) sin β p(b)u (b) = (19), (p()u ) + q()u = cos α u() sin α p()u () = cos β u(b) sin β p(b)u (b) = (2) [,b] ϕ, (19) f()ϕ()d = 513 (Fredholm ) f (19) (2) u 56 Strum-Lioville, λ (p()ϕ ) + {q() + λr()}ϕ =, (, b) (21), p, q, r [, b], p() >, r() >, cos α ϕ() sin α p()ϕ () = (22) cos α ϕ() sin α p()ϕ () = (23), λ ϕ ϕ, 73

19 , ϕ, ψ (21) (22) ( ), cos α ϕ() sin α p()ϕ () = cos α ψ() sin α p()ψ () = ( )( ) ϕ() p()ϕ () cos α ψ() p()ψ = () sin α ( ) t (cos α, sin α), p()w (ϕ, ψ)(), p()w (ϕ, ψ)() =, 54, ϕ ψ, C > ϕ = Cψ (21),(22),, (22), (23), λ ϕ λ, λ, ϕ, ϕ, ϕ λ, ϕ 55 Strum-Lioville { y + λy = y() = y(l) = 2, α 2 + λ = λ < α = ± λ y 1 () = e λ, y 2 () = e λ y() = C 1 e λ + C 2 e λ y() = C 1 + C 2 = y(l) = C 1 e λl + C 2 e λl = C 1 e λl = 74

20 C 1 = C 2 = λ < y λ =, y 1 () = 1, y 2 () =, y λ >, α = ± λi y 1 () = cos( λ), y 2 () = sin( λ) y() = C 1 cos( λ) + C 2 sin( λ) y() = C 1 = y(l) = C 1 cos( λl) + C 2 sin( λl) = C 2 sin( λl) = sin( ( nπ ) 2 λl) = λ = (n = 1, 2, ) L ( nπ ) 2 λ = λ n =, y = yn () = C n sin nπ L L 51 Strum-Liouville { y + λy = y () = y (L) =, Strum-Liouville, Strum-Lioville, 514 Strum-Liouville, λ, λ < λ 1 < λ 2 <, lim λ n =, n 1, λ n ϕ n (, b) n 75

21 515 ( ) Strum-Liouville (21),(22),(23) r(), (λ m, ϕ m ), (λ n, ϕ n ) λ m λ n r()ϕ m ()ϕ n ()d = (p()ϕ m) + q()ϕ m + λ m r()ϕ m = (p()ϕ n) + q()ϕ n + λ n r()ϕ n = 1 ϕ m, 2 ϕ n (p()ϕ m) ϕ m (p()ϕ n) ϕ n + (λ m λ n )r()ϕ m ϕ n = [, b] (p()ϕ m()) ϕ n ()d = [p()ϕ m()ϕ n ()] b (p()ϕ n()) ϕ m ()d = [p()ϕ n()ϕ m ()] b [p(){ϕ m()ϕ n () ϕ m ()ϕ n()}] b + (λ m λ n ) p()ϕ m()ϕ n()d p()ϕ n()ϕ m()d r()ϕ m ()ϕ n ()d = 1 ϕ m()ϕ n () ϕ m ()ϕ n() = cos α ϕ m () sin α p()ϕ m() = cos α ϕ n () sin α p()ϕ n() = ( )( ) ϕm () p()ϕ m() cos α ϕ n () p()ϕ = n() sin α, t (cos α, sin α),, p(){ϕ m ()ϕ n() ϕ m()ϕ n ()} = = b 76 ( )

22 λ m λ n, (λ m λ n ) r()ϕ m ()ϕ n ()d = r()ϕ m ()ϕ n ()d = 77

1 1 x y = y(x) y, y,..., y (n) : n y F (x, y, y,..., y (n) ) = 0 n F (x, y, y ) = 0 1 y(x) y y = G(x, y) y, y y + p(x)y = q(x) 1 p(x) q(

1 1 x y = y(x) y, y,..., y (n) : n y F (x, y, y,..., y (n) ) = 0 n F (x, y, y ) = 0 1 y(x) y y = G(x, y) y, y y + p(x)y = q(x) 1 p(x) q( 1 1 y = y() y, y,..., y (n) : n y F (, y, y,..., y (n) ) = 0 n F (, y, y ) = 0 1 y() 1.1 1 y y = G(, y) 1.1.1 1 y, y y + p()y = q() 1 p() q() (q() = 0) y + p()y = 0 y y + py = 0 y y = p (log y) = p log

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