D D E 2 (? 44 E E

Transcription

1 A B 23 IV 3 A A A A.2 IV ( B B B.3 Laplace B B.5 Green n C Laplace 5 C C C C.4 Fourier, Laplace C.5 Laplace C C D 27 D D.2 D D D D.3.2 e m,α D.4 (D α m u = f D.5 p(du = f

2 D D E 2 (? 44 E E E E E E E E.4.2 y + ω 2 y = E.4.3 y = E.4.4 y ω 2 y = E E E F 53 F F F G 62 G. Newton H 63 H I Kepler 66 J 66 K 66 K L 68 L M 7 M N 79 N N

3 IV A 23 IV A. 23 IV A.. 2 IV IV A..2 OK A.2 IV ( y = g (g y = f(x y = f(x y = gx + C (C, y = g 2 x2 + Cx + C (C. y = f(xg(y f(x, /g(y F, G dy g(y = f(x dx G(y = F (x + C (C y = G (F (x + C y = ay (,, y = (a byy (, 3

4 ( y = a(xy ( (2 y = a(xy + b(x ( y = ay ( y = a(xy y = a(xy + b(x 2 A. ( L[y] = y a(xy L[y + z] = L[y] + L[z], L[ky] = kl[y] L (2, ( ( (3, (4 y ( 2 (3 y + py + qy = (4 y + py + qy = f(x ( (3 λ 2 + pλ + q = 2 α, β (i α β y = Ae αx + Be βx (A, B (ii α = β y = Ae αx + Bxe αx (A, B (i α, β = a ± ib (a, b R, b y = C e ax cos bx + C 2 e ax sin bx (4 u y = u + z (z L[z] = ( L[y] = y + py + qy L[y + z] = L[y] + L[z], L[ky] = kl[y] 4

5 (a (b (c (d Laplace (e y = g ( y = ay (, y = ω 2 y ( 2 L[y] L[y + z] = L[y] + L[z] 2 or B 2 B. ( 5

6 (a 2 d y dx = A y + F (x y = e xa y + e xa x e ta F (t dt (b 2 y, y 2 y = c (xy + c 2 (xy 2 y = [c (xy + c 2(xy 2 ] + [c (xy + c 2 (xy 2]. y (5 c (xy + c 2(xy 2 = (c, c 2 y = c (xy + c 2 (xy 2, y = [c (xy + c 2(xy 2] + [c (xy + c 2 (xy 2] L[y] = y + py + qy = c (xl[y ] + c 2 (xl[y 2 ] + [c (xy + c 2(xy 2] = c (xy + c 2(xy 2. L[y] = f(x (6 c (xy + c 2(xy 2 = f(x (5, (6 c (x, c 2(x c (x, c 2 (x y = c (xy + c 2 (xy 2 B. (2 L[y] = y 6y + 8y = e x. L[z] = z = Ae 2x + Be 4x. y = c (xe 2x + c 2 (xe 4x 2 ( 6

7 y = ( c (xe 2x + c 2(xe 4x + ( c (x2e 2x + c 2 (x4e 4x (7 c (xe 2x + c 2(xe 4x = y = 2c (xe 2x + 4c 2 (xe 4x. y = ( 2c (xe 2x + 4c 2(xe 4x + ( 4c (xe 2x + 6c 2 (xe 4x. L[y] = ( 2c (xe 2x + 4c 2(xe 4x + ( 4c (xe 2x + 6c 2 (xe 4x 6 ( 2c (xe 2x + 4c 2 (xe 4x + ( c (xe 2x + c 2 (xe 4x = 2c (xe 2x + 4c 2(xe 4x. e x (8 2c (xe 2x + 4c 2(xe 4x = e x. (7, (8 ( e 2x 2e 2x e 4x 4e 4x ( c, c 2 ( c (x c 2(x = ( e 2x 2e 2x e 4x 4e 4x c (x c 2(x ( c, c 2 ( c (x 2 e x = c 2 (x 6 e 3x e x = ( = e x. 2 e x 2 e 3x. u = c (xe 2x + c 2 (xe 4x = 2 e x e 2x 6 e 3x e 4x = 3 ex L[y] = e x y = Ae 2x + Be 4x + 3 ex. 7

8 B.2 Oliver Heaviside 3 (85 925, London Devon ( p p pf(x = d dx f(x, p f(x = f(t dt ( Thomas John l Anson Bromwich ( , Wolverhampton Northampton Laplace. Laplace 2. (Jan Mikusiński Functional Analysis [32] [27] 3 Maxwell ( Maxwell Heaviside Heaviside.html 8

9 Laplace [32] ([28] ( [6] Laplace Laplace ( 4 B.3 Laplace Fourier Laplace B.4 L (n 2 y + py + qy = 4 9

10 D = d dx ( 2 ( d d y + p y + qy = dx dx D 2 y + pdy + qy = (D 2 + pd + qy = n F (λ = a j λ j j= F (Dy := n j= a j d j y dx j D 2 + pd + q { (F (D + G(Dy = F (Dy + G(Dy, (F (D G(Dy = F (D(G(Dy λ 2 + pλ + q = (λ α(λ β L[y] = (D 2 + pd + qy = ((D α(d βy = (D α((d βy L[y] = f (D α(d βy = f v := (D βy (D αv = f. v(x = C (9 v(x = Ce α(x x + x e α(x t f(t dt 5 v (D βy = v (C 5 x x =

11 y : ( y(x = C e β(x x + (9 v(t = Ce α(t x + t x e β(x t v(t dt. x e α(t s f(s ds ( ( y(x = C e β(x x + Ce α(t x + x e β(x t e α(t s f(s ds dt. x C = C = ( t u(x = e β(x t e α(t s f(s ds dt. x x x = e βx e αx f G(x = e βx e αx y = G f(x ( : x x G C = v(x = y (x βy(x, C = y(x C = C = y(x = y (x = B. α, β λ 2 + pλ + q = f I x I ( t ( u(x = e β(x t e α(t s f(s ds dt x x t y + py + qy = f(x ( u(x = u (x = x = u = e αx e βx f(x u(x = G f(x := G(x := e αx e βx G(x yf(y dy

12 (i α β G(x = e α(x y e βy dy = eαx e βx α β. (ii α = β G(x = e α(x y e αy dy = xe αx. B.2 ( α, β λ 2 + pλ + q = f I e αx e βx (α β (2 u(x := G(x yf(y dy, G(x := α β xe αx (α = β u + pu + qu = f(x, u( = u ( = G Green Green (? Green D.5. ( B. Mathematica B. ( Mathematica 6 F.3 Mathematica special[a_,b_,f_]:= Expand[Integrate[Exp[a(x-t]Integrate[Exp[b(t-s]f,{s,,t}],{t,,x}]] ( special[4,2,exp[s]] y = (2 special[2,,sin[s]] y = e 2(x t ( t e 4(x t ( t e 2(t s e s ds dt = 2 e2x + 6 e4x + 3 ex. e (t s sin s ds dt = 2 ex + 5 e2x + 3 cos x + sin x. 6 2

13 (3 special[-a,a,exp[a s]] y = e a(x t ( t (4 special[-,-,exp[-s]] y = (5 special[-,-,s^2] y = e a(t s se as ds dt = 8a 3 e ax + 8a 3 eax 4a 2 xeax + 4a x2 e ax. e (x t ( t (6 special[3,3,(s+exp[s]] y = e 3(x t ( t (7 special[3,3,cos[s]] y = e 3(x t ( t (8 special[2,,+s] y = e (x t ( t e (t s e s ds dt = 2 x2 e x. e (t s s 2 ds dt = 6e x 2xe x + x 2 4x + 6. e 3(t s (s + e s ds dt = 35 8 e3x + 8 xe3x + 9 x e 3(t s cos s ds dt = 2 25 e3x + 3 xe3x cos x 3 sin x. 5 e 2(x t ( t e (t s ( + s ds dt = 3 8 e2x x2 3 4 x. 3

14 ( f g(x = f(x tg(t dt f g f g L[y] = p (D y, D = d l dx, p(x = (x β j r j j= G j (x := (r j! xr j e β jx G := G G 2 G l, u := G f (j =, 2,, l, u L[u] = f(x B.5 Green n n y (n + a y (n + + a n y + a n y = f(x α,, α n u G(x = e α x e α 2x e α nx, u(x = G f(x u (n + a u (n + + a n u + a n u = f(x, u( = u ( = = u (n ( = Green G Laplace L[G](s = L[e αx e αnx ](s = L[e αx ](s L[e αnx ](s =. s α s α n n A j, A j = (α k α j s α j= j j k Laplace n G(x = A j e α jx G : j= G (n + a G (n + + a n G + a n G =, G( = G ( = = G (n 2 ( =, G (n ( =. 4

15 α = = α n = α G(x = xn e αx (n! Green G(x, y = { e αx e βx α β xe αx (α β (α = β G + pg + qg =, G( =, G ( = C Laplace Laplace L.Euler P.S. de Laplace ( (Euler Heaviside ( Heaviside ( Laplace [29] [6], [25] [3] Laplace ( Laplace Laplace ( C. C. (Laplace f C([, ; C L[f](s := e sx f(x dx L[f] f Laplace C. ( L[f + g](s = L[f](s + L[g](s. 5

16 C.2 ( Laplace Laplace [ ] x α L Γ(α eax = (s a. α ( (5 ( ( Laplace (2 ( Laplace L [e ax ] (s = s a L[](s = s (s > Re a, (s >. (3 ( x k Laplace [ ] x n L (s = (n! s, n (4 ( Laplace L [cos ωx] = (5 ( Laplace L [e ax xα Γ(α L [cosh ωx] = ] (s = (s ax = y [ ] L e ax xα ( y (s = e y Γ(α Γ(α s a = s s 2 + ω 2, L [sin ωx] = ω s 2 + ω 2. s s 2 ω, L [sinh ωx] = ω 2 s 2 ω. 2 e sx ax xα e Γ(α dx = Γ(α (s a α Γ(α Γ(α = (s a α. e (a sx x α dx α dy s a = (s a α Γ(α e y y α dy ( α = ( L [e ax ] (s = e sx e ax dx = e (a sx dx = [ ] e (a sx a s = ( = a s s a. (2 L[](s = e sx dx = s 6 [ e sx ] = s ( = s.

17 (3 a =, α = n L [ x k+] [ (s = e sx x k+ dx = ] ( s e sx x k+ e sx (k + x k dx s = k + L[x k ](s s 7 sx = y ( y n L [x n ] (s = e sx x n dx = e y dy s s = e y y (n+ Γ(n + dy = = n! s n+ s n+ s. n+ (4 L[cos ωx](s = e sx cos ωx dx 2 ( e sx cos ωx ( e sx sin ωx = se sx cos ωx + ( ωe sx sin ωx, = se sx sin ωx + ωe sx cos ωx ( se sx cos ωx ωe sx sin ωx ( ωe sx cos ωx + se sx sin ωx = (s 2 + ω 2 e sx cos ωx, = (ω 2 + s 2 e sx sin ωx (5 Euler Laplace L[cos ωx](s = L [ (e iωx + e iωx /2 ] = ( [ ] L e iωx (s + L [ e iωx] (s = ( 2 2 s iω + s + iω = s s 2 + ω, 2 L[sin ωx](s = L [ (e iωx e iωx /(2i ] = 2i = ω s 2 + ω. 2 ( [ ] L e iωx (s L [ e iωx] (s = ( 2i s iω s + iω ( cos ωx = Re e iωx ω L[cosh ωx](s = L [ (e ωx + e ωx /2 ] = 2 = s s 2 ω, 2 7 Γ 7 ( L [e ωx ] (s + L [ e ωx] (s = ( 2 s ω + s + ω

18 L[sinh ωx](s = L [ (e ωx e ωx /2 ] = 2 = ω s 2 ω. 2 C.3 (δ Laplace ( L [e ωx ] (s L [ e ωx] (s = ( 2 s ω s + ω L[δ](s =. Laplace C.4 (Laplace ( L [ f (n] n (s = s n L[f](s s j f (n j ( j= = s n L[f](s s n f( s n 2 f ( sf (n ( f n ( n = L[f ](s = = + s e sx f (x dx = [ e sx f(x ] ( se sx f(x dx n OK e sx f(x dx = sl[f](s. L [ f (n+] [ (f (n (s = L ] (s = sl [ ( f (n] n (s f (n ( = s s n L[f](s s j f (n j ( f (n ( = s n+ L[f](s n s j f (n+ j (. C.5 (Laplace [ ] L f(t dt (s = s L[f](s. L[F ](s = = s j= F (x = e sx F (x dx = f(t dt e sx f(x dx = s L[f](s. j= [ ] s e sx F (x s e sx F (x dx 8

19 C.6 (Laplace (2 d L[f](s = L [ xf(x] (s. ds ( n d L[f](s = L [( x n f(x] (s. ds d ds L[f](s = d ds e sx f(x dx = C.7 ( Laplace C.8 C.9 L [e αx f(x] (s = L[f g](s = L[f g](s = L[f](sL[g](s. = = e sx ( dy y e sy g(y dy = L[g](sL[f](s. [ ] L x f(x (s == e sx ( xf(x dx = L [( xf(x] (s. f(x yg(y dy dx e sx f(x yg(y dx s e st f(t dt L[f](tdt. L [e αx f(x] (s = L[f](s α. e sx e αx f(x dx = ] [ ] L [e αx x n x n = L (s α = (n! (n! e (s αx f(x dx = L[f](s α. (s α n. C. ( Laplace f T L[f](s = e st 9 T e sy f(y dy.

20 L[f](s = e sx f(x dx = n= (n+t nt e sx f(x dx. x = nt + y ( y T dx = dy, e sx = e s(nt +y = e nst e sy, f(x = f(y L[f](s = n= T e nst e sy f(y dy = T (e st n e sy f(y dy = n= e st T e sy f(y dy. C.2 C. y 5y + 6y = x + sin x + e 3x Laplace ( s 2 L[f](s sf( f ( (sl[f](s f( + 6L[f](s = s 2 + s s 3. (s 2 5s + 6L[f](s = f ( + f((s 5 + s + 2 s s 3. L[f](s L[f](s = f ( s 2 5s f( s 5 s 2 5s s 2 (s 2(s 3 + (s 2 + (s 2(s 3 + (s 2(s 3 ( 2 = f ( s 2 + ( 3 + f( s 3 s s 3 ( s + 6 s 2 4 s 2 + ( s s 3 s s 2 + s 3 ( + s 2 s 3 + (s 3 ( 2 = f ( s 2 + ( 3 + f( s 3 s s s + 6 s s s 3 + (s 3 + s + 2 s 2 + Laplace f(x = f ( ( e 3x e 2x + f( ( 3e 2x 2e 3x x + 2 e2x 7 9 e3x + xe 3x + (cos x + sin x. C. (Mathematica Mathematica Apart[] 2

21 solution=solve[(s^2-5s+6y==/s^2+/(s^2++/(s-3,y] Ly= y /. solution[[,]] Ly=Apart[Ly] Mathematica Laplace InverseLaplaceTransform[] InverseLaplaceTransform[Ly,s,x] e2x 7 9 e3x + x 6 + xe3x + (cos x + sin x ( C.2 ω R L [ e iωx] (s = s iω = C.3 L[cos ωx](s = f(x = sin ωx C.4 s + iω (s iω(s + iω = s s 2 + ω + i ω 2 s 2 + ω. 2 s s 2 + ω, L[sin ωx](s = ω 2 s 2 + ω. 2 L[f ](s = s 2 L[f](s sf( f ( ω 2 L[sin ωx](s = s 2 L[sin ωx](s s ω. (s 2 + ω 2 L[sin ωx](s = ω. L[sin ωx](s = L[f](s = f s 2 + 2s = s(s + 2 = 2 ω s 2 + ω 2. s 2 + 2s ( s s + 2 f(x = 2 ( e 2x. [ L s 2 + 2s [ ] L (x = e 2x, s + 2 ] (x = L [ s [ ] L f(t dt (s = s L[f](s ] (x = s e 2t dt = 2 ( e 2x.

22 C.5 n f n f(x = L [ s n ] (x = L [ s xn f (x f n (x = (n!. C.6 f L[f n ](s = s n ] (x = s n 2 L[f](s = ( d = ds s 2 + ω 2 s (s 2 + ω 2 2 [ ] L (t dt = x n 2s (s 2 + ω 2 2 s = ( d (s 2 + ω ds s 2 + ω 2 = d L [cos ωx] (s 2ω ds = 2ω ( d ω ds s 2 + ω 2 = [ x cos ωx 2ω L [ x cos ωx] (s = L 2ω ] (s. f n (t dt. C.7 [ ] sin ωx L (s = x = s π/2 f(x = L [sin ωx] (t dt = Arctan(s/ω x cos ωx. 2ω s ω t 2 + ω 2 dt ω ω 2 ( + tan 2 θ ω2 + ω 2 tan 2 θ dθ = π ( s ω 2 Arctan ω. C.8 L[f](s = log ( + ω2 f L [ xf(x] (s = d ( ds log + ω2 = 2ω2 s 3 s 2 + ω 2 /s = 2ω2 2 s(s 2 + ω 2 = s ( 2ω ω s 2 + ω = L [ 2ω sin ωx] (s [ 2 s x ] = L 2ω sin ωt dt (s = L [2 [cos ωt] x ] (s = L [2(cos ωx ]. s 2 22

23 xf(x = 2(cos ωx. f(x = 2( cos ωx. x C. H Heaviside { (x H(x = (x < L [H(x af(x a] = e as L[f](s. C.9 L [H(x af(x a] = = f e sx H(x af(x a dx = a e sx f(x a dx e as sy f(y dy = e as e st f(y dy = e as L[f](s. L[f](s = e as s 2 C.3 s C e s x f(x dx Lebesgue s Re s Re s e sx f(x dx ( s Re s inf σ (σ = Re s > σ =, Re s < σ =. σ ( (abscissa of convergence 23

24 Laplace ( M R ( α R ( x R f(x Me αx 8 s > α Laplace C.4 Fourier, Laplace f Fourier g Fourier Laplace F[f](y = e ix y f(x dx, F [g](x = e ix y g(y dy 2π L[f](s = e sx f(x dx F[f](y = L[f](iy (Re s > a, y = iξ iy = ξ L[f](ξ = F[f]( iξ. Laplace Fourier Fourier f(x = F [Ff] (x = 2π = e xz L[f](z dz 2πi C e ix y F[f](y dy C z = a + it ( < t <. [2] ( C.2 ( Laplace ( P (x n a,, a n [ ] L P (s (x = (s a (s a n n k= P (a k e a kx j k (a k a j. (2 P (x n a C [ ] P (s L (x = (s a n n k= P (n k (ax k e ax (n k!(k!. 8 Laplace 24

25 C.5 Laplace Schwartz [8] Yosida [3] L 2 Laplace C.3 f L 2 (, Laplace g(z := e zt f(t dt Hardy-Lebesgue H 2 ( (i g {z C; Re z > } (ii x > y g(x + iy L 2 (R g(x + iy 2 dy <. sup x> R C.4 (Paley-Wiener g H 2 ( g y g(iy L 2 (R : lim g(x + iy g(iy x R 2 dy = ( L 2. f(t := N 2π l.i.m. g(iye ity dy N N f (, f Laplace g Payley-Wiener Yosida [3] Laplace Schwartz [9] Schwartz [7] Laplace [9] [8] ([33] I 4 C.6 ( Banach X C {U(t} t Ax = lim (U(h Ix h + h X A A D(A X 25

26 A M R β > ( λ C; Re λ > β n N (λ A n M(Re λ β n {U(t} Laplace A : (λ A x = e st U(tx dt (x X, Re λ > β. s a = L [ e at] (s U(tx = lim T c+it c it e λt (λ A x dλ (c > β, t >, x D(A ({U(t} x X t A L(X (X U(t = e ta def. = U(tx = lim n e ta n x, n= U(tx = lim n A n = A t n n! An ( ( + t n A n x, ( + n A (. C.7 f(x L[f](s s s > x n n! s n+ s > e αx s α s > Re α cos ωx s s 2 + ω 2 s > sin ωx ω s 2 + ω 2 s > cosh ωx s s 2 ω 2 s > ω sinh ωx ω s 2 ω 2 s > ω 26

27 ϕ(s := L[f](s Laplace f(x ϕ(s x n f(x ( n ϕ(s x f(x ϕ(t dt s e ax f(x ϕ(s a f (x sϕ(s f( f(t dt s ϕ(s f(x ah(x a f(ax (a > e as ϕ(s ( s a ϕ a D ( a j (j =, 2,, n I f : I C y = y(x y (n + a y (n + + a n y + a n y = f(x p(x = x n + a x n + + a n x + a n, D = d dx p(dy = f(x 27

28 D. ( f(x = n j= a jx j f(dy := n a j y (j j= (f(d + g(dy = f(dy + g(dy, (f(dg(d y = f(d (g(dy (f + g(x (f g(x ( f(d + g(d := (f + g(d, f(d g(d := (f g(d ( X, Y C T : X Y { T (x + y = T (x + T (y (x, y X, T (λx = λt (x (λ C, x X T x T (x T x ( L(X, Y := {T ; T : X Y }. L(X, Y C { (T + Sx := T x + Sx (T, S L(X, Y, x X, (λt x := λ(t x (T L(X, Y, λ C, x X, L(X, Y T x = (x X T : X Y T X = Y L(X, Y L(X L(X : T, S L(X ST L(X (ST x := S(T x (x X 28

29 S T (RST = R(ST L(X C (algebra id X : X x x X I λ C T λ x := λx (x X T λ L(X T λ λ ( C L(X X id X L(X T L(X, n N {} T n T n T} T {{ T} (n T n := n (n = T n+m = T n T m, (T n m = T nm (n, m T : T n T m = T m T n. T T n T n := ( T n (n Z, n < T n n T n T n+m = T n T m, (T n m = T nm (n, m Z n T L(X, f(x = a j x j C[x] j= f(t = n a j T j j= f(t L(X f(t y = n a j (T j y. j= 29

30 X X n C n C X X X D(T X T : D(T X L(X (L(X T T D(T T + S, λt, ST (T + S(x = T x + Sx (x D(T D(S (λt (x = λ(t x (x D(T (ST (x = S(T x (x {y D(T ; T y D(S} T L(X R(T := {T x; x D(T } T T L(X S : R(T y x X ( T x = y x T T D.2 D I R X = C (I; C D. ( D D L(X Dy = y (y (C X m := C m (I; C, X := X D D(D X D.2 ( f X T f y := fy ( T f L(X f ( α e αx 3

31 D. f(x C[x], α C f(de αx = f(αe αx. D k e αx = α k e αx (k =,, 2, ( ( D.2 ( u X, α, β C (, (, (2 ( D(e αx u = e αx (D + αu. ( m N e αx (D α m (e αx u = D m u. (2 m N e αx (D β m (e αx u = (D + α β m u. ( D(e αx u = αe αx u + e αx Du = e αx (D + αu. ( ( e αx D (e αx u = (D + αu. e αx αe αx u = αu e αx (D αe αx u = Du. m [ e αx (D αe αx] m u = D m u. e αx (D α m e αx u (Cf. (P AP m = P A m P (2 m = e αx (D β(e αx u = e αx [(D α + (α β] e αx u = e αx (D αe αx u + (α βu = Du + (α β = [D + (α β] u. (Cf. P (λi AP = λi P AP. ( m 3

32 D.3 D.3. D.3 ( f, g C([, ; C f g (, convolution f g f g(x = f(x yg(y dy (x [,. ( f +g h = f +(g h D.3 ( ( ( f (g h = (f g h. (2 ( f g = g f. (3 ( f (g + g 2 = f g + f g 2, f (cg = c(f g. (4 ( (f + g h = f h + g h. (5 ( a f g = f = g =. ( a E.C.Titchmarsh Injectivity (926 (5 [3] ( f, g f g( = d x df f(x, t dt = f(x, x + (x, t dt dx a a dx D. ( f C r (I; C, g C r (I; C f g C r (I; C r (f g (r (x = f (j (g (r j (x + f (r (x yg(y dy. j= 32

33 (f g (x = f(g(x + (f g (x = f(g (x + f (g(x + f (x yg(y dy, f (x yg(y dy, (f g (3 (x = f(g (x + f (g (x + f (g(x + f (3 (x yg(y dy, D. f C k ([, ; C, g C k ([, ; C, f( = f ( = = f (k ( = (f g (r ( = (R =,,, k. x = D.4 g, g 2,, g m C([, ; C f = g g 2 g m f( = f ( = = f (m 2 ( =. m m = 2 g g 2 ( =. m f = g g m h = g g m f = h g m. h( = h ( = = h (m 3 ( =. D. f( = f ( = f (m 2 ( =. D.3.2 e m,α e m (x = xm (m! (m =, 2, e m(x = e m (x = e m l e (l m D.4 (e m,α α C, m Z x m (3 e m,α (x := (m! eαx (m (m e m,α 33

34 e m, (x = x m /(m!, e,α (x = e αx D.5 (e m,α (D α α C, m N (D α l e m,α (x = e m l,α (x (l N. e m,α(x = xm 2 (m 2! eαx + xm (m! αeαx = e m,α (x + αe m,α (x. (D αe m,α (x = e m,α (x. D.2 P = (e,α, e 2,α,, e m,α, D.6 u(x = J = (D αp = P J. m c j e j,α (x j= D.5 c j = (D α j u( (D α l u(x = e k,α ( = { (j =, 2,, m. m c j e j l+,α (x j=l (k = (k x = D.4 (D α m u = f D.2 34

35 D.7 ((D α m u = α C, m N, I R u C m (I; C (i (D α m u =. (ii (c,, c m C m s.t. u(x = m c j x j e αx. j= u = e αx v v := e αx u (D α m u = e αx (D α m u = e αx (D α m e αx v = D m v = (ii (c,, c m C m s.t. v = m c j x j j= m (c,, c m C m s.t. u = e αx c j x j. u(x = m c j e j,α (x D.6 j= c j = (D α j u( I ( D.8 ((D α m u = f α C, m N, I R f C(I; C u(x := e m,α f(x = j= (x y m e α(x y f(y dy (m! (D α m u = f, u( = u ( = = u (m ( = ( v = e αx u (D α m u = f e αx (D α m e αx v = e αx f(x D m v = e αx f(x, 35

36 u( = u ( = u (m ( = v( = v ( = v (m ( = e αx f(x m v v(x = m e αy f(y dydx dx m. F (x = e αx f(x v m F : v(x = } {{ } F (x. m v(x = xm (m! F (x = } {{ } (x = m u = e αx v u(x = xm (m! (x y m F (y dy = (m! (x y m e α(x y f(y dy. (m! (x y m e αy f(y dy. (m! ( D. ((D α m u = f I, f X = C(I; C, α C, m N u (D α m u = f (I u(x = m c j e j,α (x + e m,α f(x = j= m j= c j x j (j! eαx + (x y m e α(x y f(y dy. (m! c j = (D α j u( (j =, 2,, m. D.5 p(du = f p(x C[x] p(du = f p(x r (4 p(x = (x λ j m j (λ j, a C; j k = λ j λ k, m j j= 36

37 D.2 (e k,λj p(dy = p(x (4 (5 e k,λj (x = xk (k! eλ jx (j =, 2,, r; k =, 2,, m j p(dy = c jk y = m r j c jk e k,λj (x j= k= p(dy = j {, 2,, r} ( [ ] p(d = (D λ j m j e k,λj (k =, 2,, m j j j (D λ j mj (D λ j m j y = p(dy = D.3 (5 {e k,αj } (6 m r j c jk e k,λj (x = (x I j= k= c jk = J {, 2,, r} c Jk = (k =, 2,, m J [ ] T l := (D λ j m j (D λ J l (l =,,, m J j J (D λ J m J e k,λj (x = e k mj +,λ J (x = { e,λj (x = e λ J x (k = m J (k < m J 37

38 (6 T mj ( r m j = T mj c jk e k,λj (x = j= k= [ ] ( m j m J (D λ j m j (D λ J m J c jk e k,λj (x + c Jk e k,λj (x j J [ ] = + (D λ j m j c J,mJ e λ J x j J = j J(λ J λ j m j c J,mJ e λ J x. j J k= k= c J,mJ =. (6 j J (D λ j m j ( D.2 ( p(dy = ( p(x (4 p(dy =, D = d dx C (I; C n e k,λj (x = xk (k! eλ jx (j =, 2,, r; k =, 2,, m j y = m r j c jk e k,λj (x j= k= (c jk p(dy = D.9 ( p(dy = f p(x (4 R I f C(I; C u(x := e mr,λ r e mr,λ r e m2,λ 2 e m,λ f(x p(du = f, u( = u ( = u ( = = u (n ( = y = e m,λ f 38

39 (D λ m y = f. y 2 = e m2,λ 2 y = e m2,λ 2 e m,λ f (D λ 2 m 2 y 2 = y, (D λ m (D λ 2 m 2 y 2 = y, y j = e mj,λ j y j = e mj,λ j e m2,λ 2 e m,λ f (D λ j m j y mj = y mj, (D λ m (D λ 2 m2 (D λ j m j y j = f, y r = e mr,λ r e m2,λ 2 e m,λ f p(dy r = (D λ m (D λ 2 m2 (D λ r mr y r = f G(x := e mr,λ r e mr,λ r e m2,λ 2 e m,λ (x u(x = G f(x = G(x yf(y dy G Green Green n = 2 e αx e βx (7 e αx e βx (α β = α β xe αx (α = β. β α e αx e βx e αx e αx } e αx e αx {{ e αx } = xm (m! eαx = e m,α (x m ( G λ e λx 39

40 α j (j =, 2,, n n e αx e αnx e α jx = (α j α k. j= k j (7 α β Laplace L [e α x e αnx ] (s = n L [e αjx ] (s = j= n j= s α j. n n A j = s α j s α j j= = j= n A j (s α k s = α l [ = A l (α l α k A l = (α l α k ]. k l j= k j e α x e αnx = k l n A j e αjx. j= (Laplace Green Green p(s = r (s α j m j j= ( Heaviside Green [2] 9 9 ( [5] Green ( Green [], [2] ( [3] u = G f [] [2] ( Green Green ( 4

41 D. (Green n p(d (p(x C[x] p(dg(x =, G( = G ( = = G (n 2 ( =, G (n ( = G f C([, ; C u := G f p(du = f, u( = u ( = = u (n ( = G u(x = u (x = G(f(x + G(x yf(y dy G (x yf(y dy, u (x = G(f (x + G (f(x +. u (r (x =.. r G (j (f r j (x + j=. G (x yf(y dy, G (r (x yf(y dy, u (n (x = G(f (n 2 (x + + G (n 2 (f(x + u (n (x = G(f (n (x + + G (n (f(x + G (n (x yf(y dy, G (n (x yf(y dy. u ( = u ( = = u (n ( = G( = G ( = = G n 2 ( =. p(du = G (n (f(x + p(dg(x yf(y dy. G p(dg(x =, G (n ( = p(du = f. f p(du = f G (n ( =, p(dg(x = Green Titchmarsh injectivity theorem 4

42 D.5. 2 [2] y + py + qy = f(x, y( = y ( = Green G = G(x y y(x = G(x yf(y dy Duhamel α, β (8 G(x = eαx e βx α β α, β = a ± ib (a, b R, b ax sin bx G(x = e b α (9 G(x = xe αx = xe px/2 e αx e βx lim β α α β ( D. λ 2 + pλ + q = 2 α, β u(x := G(x yf(y dy, G(x := eαx e βx α β u u + pu + qu = f(x, u( = u ( = u( = G( = u (x = G(x xf(x + G (x yf(y dy = G (x yf(y dy J.M.C.Duhamel ( , 834 Duhamel 42

43 u ( =. G ( = u (x = G (x xf(x + G (x yf(y dy = f(x + G + pg + qg = u + pu + q = f(x + G (x yf(y dy. [G (x y + pg (x y + qg(x y] f(y dy = f(x D.2 λ 2 + pλ + q = α u(x := G(x yf(y dy, G(x := xe αx u u + pu + qu = f(x, u( = u ( = (xe αx G (9, (8 u(x = G(x yf(y dy G u( = u (x = G(f(x + G (x yf(y dy u ( = G(f(. f u ( = (2 G( = u (x = G (f(x + = u (x + pu (x + qu(x = G (f(x + G (x yf(y dy [G (x y + pg (x y + qg(x y] f(y dy. f (2 G ( =, G + pg + qg = 43

44 (2, (2 G C, C 2 { C e αx + C 2 e βx ( 2 α, β G(x = (C + C 2 xe αx ( α. u( = u ( = C, C 2 (8, (9 D.6 TO DO LIST [4] E 2 (? (22 y + py + qy = f(x. (23 y + py + qy =. E. E. ( 2 ( λ 2 + pλ + q = 2 α, β (24 y = Ae αx + Be βx (A, B (23 (a A, B (24 y (23 (b (23 A, B y = Ae αx + Be βx 44

45 E.2 ( 2 (2 λ 2 + pλ + q = α (25 y = Ae αx + Bxe βx (A, B (23 (a (25 y (23 (b (23 A, B y = Ae αx + Bxe βx 2 2 ( E.2 ( E.2. y + ω 2 y = (ω y y y + ω 2 yy =. ( 2 (y ω2 y 2 = C s.t. 2 (y ω2 y 2 = 2 ω2 C 2. y = ± ω 2 C 2 ω 2 y 2 = ±ω C 2 y 2. dy C2 y = ±ω dx. 2 45

46 Arcsin y C = ±ωx + C (C. y C = ± sin(ωx + C. y = A cos ωx + B sin ωx (A, B y y ω 2 y = y = A cosh ωx + B sinh ωx (A, B y = y = Ax + B (A, B. E.2.2 E. D = d/dx α C, m N (26 (D αy = e αx D(e αx, (27 (D α m y = e αx D m (e αx. E. y = e px/2 z y = e px/2 z y + py + qy = z + y = p 2 e px/2 + e px/2 z, y = p ( p 2 2 e px/2 z + e px/2 z + (q p2 z =. 4 ( p ( p 2 z + z e px/2 2 = 4 z pz + z e px/2 46

47 y + py + qy = ( p 2 4 z pz + z e px/2 + ] (q p2 z. 4 = e px/2 [z + 2 E. y + py + qy = D 2 y + pdy + qy = = e px/2 D 2 (e px/2 y + e px/2 (e px/2 y ( p2 2 z + pz e px/2 + qe px/2 z ( D + p 2 y + (q p2 y 2 4 y + py + qy = e px/2 D 2 (e px/2 y + e px/2 (e px/2 y =. D 2 (e px/2 y + (e px/2 y =. y + py + qy = y = e px/2 z z z + (q p2 z = 4 (i q p 2 /4 > ω = q p2 4 z + ω 2 z = z = A cos ωx + B sin ωx (A, B. (28 y = e px/2 (A cos ωx + B sin ωx (A, B. (ii q p 2 /4 = z = Ax + B z = (A, B. (29 y = e px/2 (Ax + B (A, B. 47

48 (iii q p 2 /4 < p 2 ω = 4 q z ω 2 z = z = A cosh ωx + B sinh ωx (A, B. (3 y = e px/2 (A cosh ωx + B sinh ωx (A, B. E.3 ( E.2 ( y = ay + f(x, y(x = y y = y e a(x x + x e a(x y f(y dy. y + py + qy =, y(x = y, y (x = y λ 2 + pλ + q = 2 α, β v := (D βy (D α [(D βy] =. (D αv = i.e. v = αv, v(x = y (x βy(x = y βy. (3 v(x = (y βy e α(x x. y (D βy = v i.e. y = βy, y(x = y y(x = y e β(x x + 48 x e β(x y v(y dy.

49 (3 y(x = y e β(x x + α = β x e β(x y (y βy e α(y x dy = y e β(x x + (y βy e βx αx x e (α βy dy. (x x (α = β e (α βy dy = x e e (α βx α β (α β. y = y e α(x x + (y αy e αx αx (x x = e α(x x [y + (y αy (x x ], α β y(x = y e β(x x + (y βy e βx αx e(α βx e (α βx α β = y e β(x x + y βy [ e α(x x e ] β(x x α β = y βy α β eα(x x + y αy β α eβ(x x. E.3 p, q, y, y C, x R y + py + qy =, y(x = y, y (x = y α, β : e α(x x [y + (y αy (x x ] (α = β y = y βy α β eα(x x + y αy β α eβ(x x (α β. E.4 y ω 2 y = y = y + ω 2 y = E.4. y + py + qy = 49

50 y + ω 2 y =, y =, y ω 2 y = y(x = y, y (x = y ( y = y = y E.4.2 y + ω 2 y = E(x E(x := 2 [(y 2 + ω 2 y 2 ] E (x = y y + ω 2 yy = y [ y + ω 2 y ] = y = x y = y = E(x E(x = [ (y 2 + ω 2 (y 2] = 2 y y. E.4.3 y = y = y (x x + y E.4.4 y ω 2 y = ( 5

51 E.5 E.3 (32 y + py + qy = λ 2 + pλ + q = 2 α, β ( α β e αx, e βx (32 (2 α = β e αx, xe αx (32 ( E.4 y + py + qy =, y(x = y, y (x = y u := y, u 2 := y, u := (u, u 2 T ( ( d dx u = u = u 2 y y u(x = = ( ( y py qy u (x u 2 (x = ( = ( y(x y (x u 2 pu 2 qu = ( y y = ( q p u, A := ( q p u, u := ( y y d dx u = A u, u(x = u. u(x = u + x A u(y dy u = u(x, v = v(x u(x v(x = x A ( u(y v(y dy, u(x v(x = w(x := u(x v(x w(x = A w(y dy, w(x =. x 5

52 x I n N M := max w(x x I w(x A M x x n n n! w(x = (x I. I w. u = v. E.6 ( E.5 E.6 E.7 f(de λt = f(λe λt (f + g(d = f(d + g(d, (f g(d = f(dg(d. D j (e λx u = e λx (D + λ j u f(d ( e λx u(x = e λx f(d + λu(x. e λx f(de λx = f(d + λ. α β e αx e βx C e αx + C 2 e βx = D α C 2 (β αe βx =. C 2 =. C = C = C 2 = α = β e αx xe αx C e αx + C 2 xe αx = D α C 2 (β αe βx =. C 2 =. C = C = C 2 = 52

53 E.7 n F 24 IV ( F ( 27 2 ( x 3 y + y 2 = (2 y = 3y 2/3 (3 y = y (4 x 2 y + y 2 = (5 y 3 + x 6 y = (6 y xy = x 2 y (7 y + ay 2 = (8 sin x sin 2 y y cos x = (9 ( + xy + ( yxy = ( y tan x = cot y ( ( + x 3 y + x 2 y 2 = (2 y = a(b 2 y 2 (3 y = cos2 y + x (4 2 y = + sin x (5 y = xy sec 2 x x 2 (6 x( + y2 y = y( + x 2 (7 yy = x(y + (8 xy y 2 + = (9 y = e 2(x+y (2 y = e (x+y (2 y = y (22 y = x y (23 y = x (24 y = y (25 y = y2 x (26 2 y = y2 x (27 3 y = x + y 2 y + x 2 ( dy y = dx 2 x 3 y = 2 x 2 + C = 2Cx2. 2x 2 y x 2 y = Cx 2 /2. (2 y 2/3 dy = 3dx y x 3 3y /3 = 3x + 3C y /3 = x + C y = (x + C 3. (3 (y /2 dy = dx 2 [ ] [2] 53

54 2(y /2 = x + C. y (4 y (5 y = + (x + C2. 4 dy y 2 = x 2 dx y = x + C = Cx x y = x Cx. y 3 dy = x 6 dx y 2 y 2 = 5 x 5 + C y = ( 2 C. 5x 5 p.2 y 2 = 5x2 Cx 5 2 y 2 = 5x5 Cx 5 2 (6 dy/dx log y = dx x(x + = y = (x + x 2 dy dx dy y = dx x + x 2 ( x dx = log x log x + + log C = log C x + y = C x x +. x x + 54

55 (8 y dy sin 2 y = tan x dx cot y = log cos x log C y = Arccot (log C cos x (9 ( ( dy = + dx y x y log y = x + log x + C. F (y = y log y F < y < y > ( tan y dy = cot x dx log cos y = log sin x + log C = C sin x cos y cos y = C sin x ( C y = Arccos. sin x F.2 a, b, c, d ( y + ay = (2 y + ay = b (3 y + y cot x = cosec x ( < x < π/2 (4 y + 2xy = x (5 y y tan x = sin x ( π/2 < x < π/2 (6 y 2xy = e x2 (7 xy + y = x log x (x > (8 y + ay = e bx (9 y + a x y = ( y xy = x ( y + x y = x2 (x > (2 xy + y = 4x( + x 2 (3 xy (y + x 2 sin 2 x = (4 y + y cos x = sin xe sin x (5 x( x 2 y + (x 2 y = x 3 ( < x < (6 y ay = sin x (7 ( + x 2 y = xy + x 2 (8 y + ( + x 2 y = e x3 /3 (9 y + ay = bx 2 + cx + d (2 xy + ( + xy = e x ( y = Ce ax (2 y = ay + b y = ay y = Ce ax (C y = c(xe ax y = c (x e ax + c(x e ax ( a = ay + c (xe ax. c (xe ax = b. 55

56 c (x = be ax c(x = a b eax + C (C. y = c(xe ax = Ce ax + a b. (3 y = (cot xy + sin x y = (cot xy dy y = ( cot x dx = log sin x + log C = log sin x + log C = log C sin x y = ± C sin x = C sin x y = c(x sin x y = c (x cos x + c(x sin x sin 2 x = (C. y sin x + c (x sin x. c (x sin x = sin x. c (x =. c(x = x + C (C. y = c(x sin x = x + C sin x. (4 y = 2xy + x y = 2xy dy y = ( 2x dx = x 2 + C (C y = ±e C e x2 = C e x2 (C. y = c(xe x2 y = c (x e x2 + c(x e x2 ( 2x = 2xy + c (xe x2. c (x = xe x2. c (xe x2 = x. c(x = ex2 2 + C (C. ( y = c(xe x2 = 2 ex2 + C e x2 = C e x

57 (5 y = 2xy + e x2 y = 2xy dy y = 2x dx = x 2 + C (C. y = ±e C e x2 = C e x2 (C. y = c(xe x2 y = c (x e x2 + c(x e x2 (2x = 2xy + c (xe x2. c (xe x2 = e x2. c (x =. c(x = x + C (C. y = c(xe x2 = (x + C e x2. (6 y = 2xy + e x2 y = 2xy dy y = 2x dx = x 2 + C (C. y = c(xe x2 y = ±e C e x2 = C e x2 (C. y = c (x e x2 + c(x e x2 (2x = 2xy + c (xe x2. c (xe x2 = e x2. c (x =. c(x = x + C (C. (7 y = y x y = c(xe x2 = (x + C e x2. + log x y = y x dy dx y = x = log x + log C = log C x (C. y = ± C x = C x y = c(x/x y = c (x x + c(x 57 (C. ( x 2 = y x + c (x x.

58 c (x x = log x. c (x = x log x. ( x 2 c(x = x log x dx = log x dx = x2 log x x y = c(x x = x log x 2 x 4 + C x. x dx = x2 log x 2 x2 4 +C (C (8 y = ay + e bx y = ay y = Ce ax (C y = c(xe ax y = c (x e ax + c(x e ax ( a = ay + c (xe ax. c (xe ax = e bx. (9 c (x = e (a+bx. c(x = e(a+bx a + b + C (C. y = c(xe ax = ebx a + b + C e ax. ( ( y = y x + ( x2 y = y/x ( dy y = Dx = log x + log C = log C x x y = ± C x = C x y = c(x/x y = c (x x + c(x (C. ( x 2 = y x + c (x x. (C c (x = x x 3. c(x = x2 y = c(x x c (x x = x2. 2 x4 + 4 C (C. = x 2 x3 4 + C x. 58

59 (2 (3 (4 y = y cos x sin xe sin x y = y cos x dy y = ( cos x Dx = sin x + C (C y = ±e C e sin x = C e sin x (C. y = c(xe sin x y = c (x e sin x + c(x e sin x ( cos x = (cos xy + c (xe sin x. c (xe sin x = sin xe sin x. c (x = sin x. c(x = cos x + C (C. y = c(xe sin x = C e sin x + e sin x cos x. (5 (6 (7 y /y = x/ + x 2 dy y = x dx = + x 2 + C (C. + x 2 y = ±e C exp + x 2 = C exp + x 2 (C. (8 (9 (2 y = + x x y + ex x y = ( + xy/x ( dy y = + Dx x log y = (x + log x + log C = log y = ± C xe x = C xe x C x e x (C. (C. 59

60 y = c(x xe x y = c (x + x + c(x = xex x y + c (x xe. x c (x xe x = ex x. c (x = e 2x. c(x = 2 e2x + C (C. y = c(x xe x = C xe x + ex 2x. F.3 2 ( y 6y + 8y = e x. L[y] = y 6y + 8y L[z] = λ 2 6λ + 8 = λ = 2, 4 z = Ae 2x + Be 4x (A, B. L[y] = e x u u = Ce x (C L[u] = L[Ce x ] = (C 6C + 8Ce x = 3Ce x u 3C = C = 3. L[y] = ex y = u + z = 3 ex + Ae 2x + Be 4x. (2 y 3y + 2y = sin x. L[y] = y 3y + 2y L[z] = λ 2 3λ+2 = λ =, 2 z = Ae x +Be 2x (A, B. L[y] = sin x u u = k cos x + l sin x (k, l L[u] = L[k cos x+l sin x] = ( k cos x l sin x 3( k sin x+l cos x+2(k cos x+l sin x = (k 3l cos x + (3k + l sin x u k 3l =, 3k + l =. k = 3, l =. u = 3 cos x + y = u + z = 3 cos x + sin x + Aex + Be 2x. sin x. L[y] = sin x (3 y a 2 y = xe ax. L[y] = y a 2 y L[z] = λ 2 a 2 = λ = ±a. a a =. (i a 2 z = Ae ax + Be ax (A, B. L[y] = xe ax u = (px 2 + qxe ax 3 L[u] = L[(px 2 + qxe ax ] = ( = [4apx + 2(aq + p] e ax. xe ax 4ap =, aq + p =. 3 a e ax u = pe ax x u = (px + qe ax a x u = (px 2 + qxe ax 6

61 p = y = u + z = 4a, q = ( 4a. u = 2 4a x2 4a x ( 2 4a x2 4a x e ax + Ae ax + Be ax. 2 e ax. L[y] = xe ax (ii a = ±a = ( z = Ae x + Bxe x = A + Bx (A, B. a = L[y] = xe ax y = x u = x3 6 L[y] = xe ax y = u+z = x3 6 +A+Bx. (4 y + 2y + y = e x. L[y] = y + 2y + y L[z] = λ 2 + 2λ + = λ = ( z = Ae x + Bxe x (A, B. L[y] = e x u u = kx 2 e x (k 4 L[u] = L[kx 2 e x ] = ( = 2ke x u 2k = k = 2. y = u + z = 2 x2 e x + Ae x + Bxe x. u = 2 x2 e x. L[y] = e x (5 y +2y +y = x 2. L[y] = y +2y +y ( L[z] = z = Ae x +Bxe x (A, B. L[y] = x 2 u u = px 2 +qx+r (p, q, r 5 L[u] = L[px 2 +qx+r] = ( = px 2 + (4p + qx + (2p + 2q + r u 2k = k = 2. u = 2 x2 e x. L[y] = e x y = u + z = 2 x2 e x + Ae x + Bxe x. (6 y 6y + 9y = x + e x. L[y] = y 6y + 9y L[z] = λ 2 6λ + 9 = λ = 3 ( z = Ae 3x + Bxe 3x (A, B. L[y] = x u u = px + q (p, q L[u] = L[px + q] = ( = 9px + (9q 6p u 9p =, 9q 6p =. p = /9, q = 2/27. u = 9 x L[y] = e x v u = ke x (k L[u] = L[ke x ] = ( = 4ke x u 4k = k = 4. u = 4 ex. L[y] = x + e x y = u + v + z = 9 x ex + Ae x + Bxe x. (7 y 6y + 9y = cos x. L[y] = y 6y + 9y L[z] = λ 2 6λ + 9 = λ = 3 ( z = Ae 3x + Bxe 3x 4 u = ke x x

62 (A, B. L[y] = x u u = k cos x + l sin x (k, l L[u] = L[k cos x + l sin x] = ( = (8k 6l cos x + (8l 6k sin x u 8k 6l =, 6k + 8l =. k = 2 25, l = u = 2 25 x L[y] = cos x y = u + z = 2 25 x Ae3x + Bxe 3x. (8 y 2y = + x. L[y] = y 2y L[z] = λ 2 2λ = λ =, 2 z = Ae 2x + B (A, B. L[y] = x u u = (kx + lx (k, l 6 L[u] = L[kx 2 + lx] = ( = 2kx + (2k l u 2k =, 2k l =. k =, l = 2. 2 u = 2 x2 2x. L[y] = cos x y = u + z = 2 x2 2x + Ae 2x + B. G ( 23 II [23] [8] ( G. Newton Newton (Isaac Newton, ( ( 7 (, 687 Kepler 8 6 u = kx + l x u = (kx + lx 7 [3] 8 Yohannes Kepler (57 63 Ticho Brahe (546 6 Brahe Kepler

63 9 ( 2 Gamov [5], [], Arnold [] [5] Kepler [] [] Newton Leibniz [7] Newton 676 ( Leibniz (Gottfried Wilhelm Leibniz, aaaaaa cc d æ eeeeeeeeeeeeee ff iiiiiii lll nnnnnnnnn oooo qqqq rr ssss ttttttttt vvvvvvvvvvvv x Data æqvatione qvotcvnqve flventes qvantitates involvente flvxiones invenire, et vice versa. Newton (? ( H Mathematica, Maple Mathematica DSolve H. ( x 3 y + y 2 = 9 (Kepler Kepler 63

64 DSolve[x^3 y [x]+y[x]^2==,y,x] 2 2 # Out[3]= {{y -> ( & }} # C[] y = 2x2 cx 2 (2, y = 3y 2/3 DSolve[y [x]==3 (y[x]^(2/3,y,x] Out[4]= {{y -> (# - 3 # C[] + 3 # C[] - C[] & }} y = x 3 3Cx 2 + 3C 2 x C 3 = (x C 3 (3, y = y DSolve[y [x]==sqrt[y[x]-],y] # - 2 # C[] + C[] Out[5]= {{y -> ( & }} 4 y = 4 + x2 2Cx + C 2 4 = (x C = + (x C2 4 (4 x 2 y + y 2 = 64

65 DSolve[x^2 y [x]+y[x]^2==,y,x] # Out[6]= {{y -> ( & }} - + # C[] y = x Cx (6 y xy = x 2 y DSolve[y[x]-x y [x]==x^2 y [x],y,x] Log[#] - Log[ + #] Out[7]= {{y -> (E C[] & }} y = Ce log x log(+x = Cx x + (9 ( + xy + ( yxy = DSolve[(+xy[x]+(-y[x]x y [x]==,y,x] InverseFunction::ifun: Inverse functions are being used. Values may be lost for multivalued inverses. Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found. -# - C[] E Out[8]= {{y -> (-ProductLog[-( ] & }} # y log y = x + log x + C (C 65

66 y ( y tan x = cot y DSolve[y [x]tan[x]==cot[y[x]],y,x] y = Arccos C sin x (C I Kepler ( [2] ( [4] [3] J,, (. K Lipschitz Picard Picard Lindölef Cauchy K. K. dy dx = f(x, y y(x = y (x I, y = ϕ (x (x x x, y = ϕ 2 (x (x x x 2 ϕ (x = ϕ 2 (x (x x x := min{x, x 2 }. 66

67 ϕ, ϕ 2 ϕ j (x = y + x f(t, ϕ j (t dt (j =, 2 ϕ(x = ϕ (x ϕ 2 (x ϕ(x ϕ(x = x f(t, ϕ (t f(t, ϕ 2 (t dt M = max x [x,x ] ϕ(x ϕ(x LM(x x, ϕ(x L ϕ(x L x [f(t, ϕ (t f(t, ϕ 2 (t]. x L ϕ (t ϕ 2 (t dt = L LM(t x dt = L 2 M (x x 2, x 2 L 2 M (t x 2 dt = L 3 M (x x 3, x 2 3! x ϕ(t dt. ϕ(x M [L(x x ] n. n! ϕ(x M [L(x x ] n. n! ϕ. ϕ ϕ 2. K.2 ( Lipschitz dy dx = f(x, y (x I, y(x = y y = ϕ (x (x x x, y = ϕ 2 (x (x x x 2 ϕ (x = ϕ 2 (x (x x x + := min{x, x 2 }. x := sup E, E := {β [x, x + ]; x [x, β] ϕ (x = ϕ 2 (x} x E E x + E E sup E 67

68 : ϕ = ϕ 2 on [x, x] ϕ = ϕ 2 on [x, x x [x, x sup x (x, x E. ϕ = ϕ 2 on [x, x ] ϕ (x = ϕ 2 (x. ϕ (x = ϕ 2 (x x n x {x n } ϕ (x n = ϕ 2 (x n n ϕ, ϕ 2 ϕ (x = ϕ 2 (x. 2 x = x ϕ = ϕ 2 on [x, x ]. x E x sup E x. x < x y := ϕ (x = ϕ 2 (x ϕ, ϕ 2 dy dx = f(x, y (x (x, x, y(x = y < x x + s.t. ϕ = ϕ 2 on [x, x]. x < x, x E x = sup E L L. ( ( n sin n (2 cos n n 2 (3 (4 n= n= n n 2 π (5 ( n n n= n= n= n ( ( n 2 2n + x2n+ R x < R x n= f(x = n= ( n 2n + x2n+ ( R (2 f (x (3 f(x (4 lim f(x x R ( y, y dy dx, d2 y dx 2 3 ( y = xy (2 y = xy + x x = y = 68

69 4 (a y + 2y 3y =, (b y + 2y 3y = e x ( (a (2 (a y( =, y ( = 2 lim (3 (b x y 69

70 M 5 ( lim x sin x x (2 (3 = lim n n sin n = M. ( a n n lim a n =. lim a n = a n n n n n, n= n = =. n n= M.2 ( n a n b n b n a n n n a n b n n n : a n b n a n b n n n <, = a { a x α dx = (α > α (α cos n n 2 n 2, n= n 2 cos n n 2 M.3 a n ( a n a n n n n n 7

71 M.4 ( n a n b n b n n a n ( a n n M.5 ( α n= n α = { (α > (α. : n n 2 n= (4 a n = n2 π n n a n+ a n ( (n + 2 n 2 = π n+ π = + 2 n n π π < an a n+ M.6 (d Alembert (ratio test lim n a n a n n r < = ( r > =. n n= = r r = ( ( r n n= r < r (5 a n = ( n n n a n = n a n an 7

72 M.7 ( (Leibniz a n a a 2 a 3 (n a n n 2 ( ( ( x ( n ( ( x 2 n 2n + n= ( n ( 2n + y2 R a n+ lim n a n n= 2n + = lim n 2(n + + R =. a n = ( n 2n + 2n = lim n 2n + 3 = lim n n n = 2 2 = = R y < = (, y > = (. x < y <, x > y > R =. x < = (, x > = (. (2 ( x < R = f ( n (x = 2n + (2n + x2n = ( x 2 n. n= x 2 x < x 2 < f (x = ( x 2 = + x n=

73 (3 f( = f(x = f( + f (t dt = + (4 f(x = Arctan x x = + t dt = [Arctan 2 x]x = Arctan x. lim f(x = lim Arctan x = Arctan = π x R o x 4. 3 ( ( dy dx = xy log y = x2 2 dy y = x dx. + C (C. ( x 2 y = exp 2 + C = e C exp ( ( x y = ±e C 2 x exp = C 2 exp 2 2 ( x 2 (2 y = C(xe x2 /2 dy dx = C (xe x2 /2 + C(x 2. (C ( x 2 e x2 /2 = xc(xe x2 /2 + C (xe x2 /2 = xy + C (xe x2 /2. 2 C(x = C (xe x2 /2 = x. C (x = xe x2 /2. xe x2 /2 dx = e x2 /2 + C (C. y = C(xe x2 /2 = ( e x2 /2 + C e x2 /2 = + C e x2 /2. 73

74 4 ( λ 2 + 2λ 3 = λ =, 3. (a y = Ae x + Be 3x (A, B. (2 y( = A + B =, y ( = 2 A 3B = 2 A = 5/4, B = /4. y = 4 ex 5 4 e 3x. lim y =. x (3 e x ( ( n =, α =, m = u = px m e αx = pxe x u + 2u 3u = 4pe x. e x 4p =. p = /4. u = 4 xex (b y = Ae x + Be 3x + 4 xex (A, B. f(x y + py + qy = f(x. f(x = n e αx α m (m u(x = (n x m e αx L[u] = { } 2. f(x = (n e ax cos bx a + ib m (m sin bx u(x = n x m e ax (A cos bx + B sin bx M. ( ( cos n (2 n 4 3 (3 + sin n (4 ( n sin n n 2 n n= n= n= n= 74

75 2 n xn R x < R x n= f(x = n= n xn ( R (2 f (x (3 f(x (4 lim f(x x R+ ( y, y dy dx, d2 y dx 2 3 ( y = x y (2 y = x y + ex x = y = 4 (a y + y + y =, (b y + y + y = + x ( (a x y (2 (a y( =, y ( = (3 (b 75

76 23 IV ( lim cos n n = (2 a n = n4 3 n lim a n+ a n = lim n n (n n+ n 4 3 n ( (n + 4 = lim 3n n 3 n+ n = lim n n 3 = 3 = 3. an (3 + sin n n 2 + = 2 n 2 n 2 2 n + sin n 2 n 2 n= n= (4 < n n y = sin x x n sin n lim sin n n = < sin n ( (i, (ii sin n, (iii lim sin n n = 3 ( ( (2, (3 (4 ( 2 ( x n a n : a n = n. lim a n+ n a n = lim n n n + = lim n + n =. /R R =. (2 ( x < R = f (x = n nxn = x n = x n. n=, x = x < (3 f( = n= f (x = x. n n = n= f(x = f( + f (t dt = + n= x t dt = dt = log x = log( x. t 76

77 (4 lim f(x = lim ( log( x = log [ ( ] = log 2. x R+ x + ( a n x a n (2 ( (3 dt = log t ( t t dt = dt = log t t dy 3 ( y = x C dx log y = log x + log C = log x ( C. y = C x. (2 y = c(x x y = ±C x = C x y = c (x x c(x x 2 (C. = y x + c (x x. y y = y x + ex c (x x = ex. c (x = xe x c(x = xe x dx = xe x (x e x dx = xe x e x dx = xe x e x + C (C. y = c(x x = ex ex x + C x. x = y = y = e x ex x. = + C = + C = C C =. 77

78 ( log y = log x + C y = e log x +C = e C e log x = C log x e e log x (a x 2 a 2 x (a x 2 (2 4 ( λ 2 + λ + = λ = ± = ± 3 2 = ± 3i. 2 3x 3x y = Ae x/2 cos + Be x/2 sin 2 2 (A, B. cos, sin e x/2 (x lim y =. x (2 y( = = A + B = A. ( 3x y = e x/2 cos A ( 3B 3x 3A e x/2 sin B y ( = ( = A 2 + 3B 2 + = A 2 + 3B 2. A = B = 3. 3x y = e x/2 cos + 3x 3e x/2 sin 2 2. (3 u = px + q (p, q u = p, u = u + u + u = + p + (px + q = px + (p + q. x + p =, p + q =. p =, q =. u = x. 3x 3x y = Ae x/2 cos + Be x/2 sin + x (A, B

79 ( 2 ( y = Ae + 3i 2 x + Be 3i x 2 cos, sin ( cos, sin lim (lim (2 ( (3 ( ( (3 ( N ( [7] ( ( [] [3] [5] ( 2 N. N.. 2 ( 79

80 ( [2] (2 [] (999 ( [25] ( = [4] (982, [5] (988, [2] (99, [24] (963, [26] (993 ( [9] ( 2/3 [] V I,, (999. [2],, (2. [3],, (99. [4],, (, 982. [5] (George Gamow (,, (977. [6], Fourier-Laplace,, (993. [7] L. (L.Schwartz,,,,, (97. [8] L. (L.Schwartz,,,, (966. [9] Laurent Schwartz, Transformation de Laplace des distributions, Comm.Sém.Math. de l Univ. de Lund, tome suppl. dédié à M.Riesz, (952. [],, (999. [] (,, (998. 8

81 [2], [ ], (984. [3] (23.,, [4],, (994. [5],, (988. [6],, (989. [7], (, G5, (96. [8],,,, (97. [9],,, (997. [2],, (99. ( V [2] (2. [22],, (99. [23] E.T.,,,,, (976. [24] (L.S.Pontryagin (,,, (963. [25], 5, (997. [26], I,, (993., (23 [27] (Jan Mikusiński,,, (965. (Jan Mikusiński,,, (967. [28],, (984. [29], II, (957. [3],, (957. [3] Kôsaku Yosida, Functional analysis, sixth edition, Springer (98. [32],, (982. 8

82 [33],,,,,,, 4, (976. [34] E.C.Titchmarsh, The zeros of certain integral functions, Proceedings London Mathematical Society 25 (926,