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1 24 3, ( ) , , (Bernoulli) (Riccati) () , (2) ( ) (3) 32

2 A 4 A A B 2 4 B B B.3 Laplace B B.5 Green n C 45 D 23 IV 45 D D D D.2 IV ( ) E 2 48 E E E.3 Laplace E E.5 Green n F Laplace 56 F F F F.4 Fourier, Laplace F.5 Laplace F F G 67 G G.2 D G G G.3.2 e m,α G.4 (D α) m u = f

3 G.5 p(d)u = f G G H 2 (?) 82 H H H H H H H H.4.2 y + ω 2 y = H.4.3 y = H.4.4 y ω 2 y = H H H I 9 I I I J 99 J. Newton K K L Kepler 2 M 2 N 3 N O 4 O P 6 P Q 4 Q Q

4 IV ( ) exp x e x log x x (y = log x x = e y ) cot x, sec x, cosec x = x, x = x x = cos x sin x, x cos sin sin x, cos, tan sin, cos, tan ( arcsin x, arccos x, arctan x ) Arcsin x, Arccos x, Arctan x sin x, cos x, tan x ([ π/2, π/2], [, π], ( π/2, π/2) ) y x y(x) =. ( ) ( ). 2. ( ) (a) ( ) (b) (c) 4

5 (differential equation) 2 ( Kepler ) (,, ),,,... t h(t) h (t), h (t) ( ) 3 () h (t) = g g (MKS g 9.8m/s 2 ) () t (2) h (t) = h (t)dt = gt + C. C (3) h(t) = h (t)dt = 2 gt2 + C t + C 2. C 2 (2) t = C = h (). C v C = h () = v. (3) t = C 2 = h(). C 2 h h(t) = 2 gt2 + v t + h. 2 (Sir Isaac Newton, ) 3 5

6 h(t) (4) h (t) = g h(t) = 2 gt2 + v t + h ( h(t)) ( 2 h (t)) (4) h(t) ( (v =, h = ) (Galileo-Galilei 4, ) )..2 ( ) x (t) = ω 2 x(t) (ω ) t x(t) ( ) x (t) x (t)x (t) + ω 2 x(t)x (t) =. { d [ x (t) 2 + ω 2 x(t) 2]} dt 2 { d [ x (t) 2 + ω 2 x(t) 2]} =. dt 2 [ x (t) 2 + ω 2 x(t) 2] = C (C ). 2 dx/dt = x (t) dx dt = ± 2C ω 2 x 2. 2C ω 2 x 2 t dx 2C ω2 x 2 dt dt = ± dt = ±t + C (C ). dx 2C ω2 x 2 2C/ω = a 2C ω 2 x 2 = ω a 2 x 2 dx 2C ω2 x = dx 2 ω a2 x = ( x ) 2 ω sin + C 2 (C 2 ). a 4 6

7 ( x ) ω sin = ±t + (C C 2 ) = ±(t + C 3 ) ( C 3 = ±(C C 2 ) ) a ( x ) sin = ±(ωt + C 3 ) (C 3 ). a x (5) x = a sin (±(ωt + C 3 )) = ±a sin(ωt + C 3 ) = C 4 sin(ωt + C 3 ). C 3, C 4 ( ) (5) x t T = 2π/ω.2 5 (ordinary differential equation), 2 (partial differential equation) ( ) ( ) (order). y = g 2 2 t, h, v x y y, y,, y (n), dy dx, d2 y dx,, dn y 2 dx, n n F (x, y, y,, y (n) ) = (F ) y (n) = G(x, y, y,, y (n ) ) (G ) 5 7

8 (solution) (solve) 2 (A) ( ) (B) (A) y = dy dx = f(x) f(x) dx ( ) ( 6 ) ( ) (B).2 y = y = y = x + C y = C x + C 2 (C ) (C, C 2 ) (C C, C 2 ) ( ) 7 y = x + C y = C x + C 2, ( Kepler ) ( ) 7 8

9 .3 ( ) (y ) 2 = 4y y = (x C) 2 (C ) y = ( ) y = { (x C) (x C) 2 (x > C) ( ) n ( ).4 h (t) = g (6) h() = h, h () = v (6) n (7) F (x, y, y,, y (n) ) = (8) y(x ) = y, y (x ) = y,, y (i) (x ) = y i,, y (n ) = y n ( x, y, y,, y n ) (7), (8) y (8) n (7), (8) { y = f(x, y) y(x ) = y x, y f(x, y) (x, y ) 8. () x 2 y + yy = 3x (2) y 2 y 2 log( + x 2 ) = (3) yy = y 2 (4) ( x 2 )y 2xy + 6y = (5) 3y (4) 2y (3) + y + e x = (6) x 2 y + xy + (x 2 )y = (7) y 2 = k( + y 2 ), k (8) (x 2 y ) + 4x 2 y = 8 [ ] [4] p.88 89, 4 9

10 d 2. f(x) = f(x) = C (C ) dx (a, b ) () xy + y = (2) xy + y x = (3) x + a 2 yy = (4) x 2 y + 4xy + 2y = (5) (x + y)( + y ) = x (6) y = 2 sec y π + x (7) yy 2 a2 y = ± (8) y 2 a2 y = ± 2 (9) y (y + y) = () xy y = x 2 f(x) (f(x) ) 3. y = f(x), y(x ) = y y = y + f(t) dt x ( ) 2 y = F (x, y) x y (9) y = f(x)g(y) ( ) 2. () dy dx = f(x)g(y) g(y) x dy g(y) dx = f(x). dy g(y) dx dx = f(x) dx. y () g(y) dy = f(x) dx. () () dy dx = f(x)g(y) dy dy g(y) = f(x)dx g(y) = f(x)dx 2. g(y)

11 2.2 x dy g(y) dx dx = x f(x) dx y(x ) y(x ) dy g(y) = x f(x) dx. () G(y) = F (x) + C (G(y), F (x), f(x), C ). g(y) y y = G (F (x) + C) (G G ) ( y = ay) (2) dy y (3) = 2 dx dy dx = 2y. dy y = 2 dx. log y = 2x + C (C ). 9 y = e 2x+C = e C e 2x y = ±e C e 2x. ±e C C y = Ce 2x (C ). (3) y (2) y = ( ) y = Ce 2x C = (4) y = Ce 2x (C ) y y 2y = 9 e x = y x = log y. d ( ) ye 2x = y e 2x + y ( 2)e 2x = e 2x (y 2y) = e 2x = dx

12 C ye 2x = C. y = Ce 2x. y() = y y = y() = Ce 2 = Ce = C. y = y e 2x. 2.4 (4) y y = ( ) y y = y y y y = y = y ( ) y x ( ) (5) y = 3y 2/3 = 3 ( 3 y) 2 ( ) y = (x C) 3 (C ) y = ( ) y = { (x C ) 3 (x < C ) (x C ), y = { (x < C 2 ) (x C 2 ) 3 (C 2 x), (x C ) 3 (x < C ) y = (C x C 2 ) (x C 2 ) 3 (C 2 < x) ( ) 2.3 2

13 2.5 a (6) dy dx = ay (7) y = Ce ax (C ) ( (6) C (7) C (7) y (6) ) y() = y y = y e ax. (6) ( ) 2.6 ( ) P. F. Verhulst (84 849) (logistic equation) (8) dy dx = (a by)y (a, b ) dy dx = ay y (8) 2 dy (a by)y = dx (a by)y = ( ) a y y a/b ( ) y dy = a y a/b dx. y = a( by)y T. R. Malthus ( ) 798 ( = ) (= = ) 2 [25] ( ) a ( ), b ( ) 3

14 log y y a/b = ax + C y y a/b = C e ax (C ). (C ). y = C e ax (y a/b). y y = a b C e ax C e ax. 3 x = y = y y = C = y y a/b. ay by + (a by )e ax. y a/b < x < y < x < y > a/b x > y = y, y = a/b y a/b (, a/b ( ) ) < y lim x y = a/b ( a/b ) a = b = dy dx = ( y)y, y() = y y = y y + ( y )e x. x, ( ) y ( ) y= PSfrag replacements y = y = y= 3 y a/b 4

15 4. () 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) ( + x)y + ( y)xy = () y tan x = cot y () ( + x 3 )y + x 2 y 2 = (2) y = a(b 2 y 2 ) (3) y = cos2 y + x 2 (4) y = + sin x sec 2 y (5) y = xy 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 y = y (25) y = y2 x (26) 2 y = y2 x (27) 3 y = x + y 2 y + x 2 y x (24) 2. () (4x + 2xy 2 )dx (x 2 y + y)dy = (2) 2y dx + e 2x dy = (3) (y 2 + )dx x dy = (4) sin 2 y dx + cos 2 x dy = (5) 4y dx + x 3 (2 + y 2 )dy = (6) y cos x dx + sin x dy = (7) 4x(y 2 + )dx y(x 2 + 2)dy = (8) e x dx e y dy = (9) ( y + y ) dx = ( x + x ) dy 3, dy = F (x, y) dx F (x, y) y a(x)y + b(x) (9) y = a(x)y + b(x) b(x) (2) y = a(x)y 3. ( ) y = a(x)y 2 y = a(x) sin y 3. (2) dy y = a(x) dx a(x) A(x) dy y = a(x) dx. log y = A(x) + C (C ). 4 [4] p.95 4, 5 5

16 y = e A(x)+C = e C e A(x) y = ±e C e A(x). ±e C C y = C e A(x). ( ) = 3.2 ( ) (2) y = a(x)y A(x) a(x) (A (x) = a(x)) y = Ce A(x) (C ) (22) y() = y (23) y = y e A(x), A(x) = a(t) dt. 3.3 (23) e exp : ( ) y = y exp a(t) dt. A(x) a(x) d dx (ye A(x) ) = y e A(x) + y e A(x) ( A (x)) = e A(x) (y A (x)y) = e A(x) (y a(x)y). y = a(x)y d dx (ye A(x) ) = ye A(x) = C (C ) y = Ce A(x) (C ). y() = x x =, y = y y = Ce A() C = y e A(). y = y e A(x) A(). a(x) A(x) A() = (2), (22) A(x) = y = y e A(x), A(x) = 6 a(t) dt a(t) dt.

17 x y(x ) = y y 3.2 (9) (variation of parameter, variation of constants) ( ) (2) y = Ce A(x) (C, A(x) a(x) ) (9) (24) y = C(x)e A(x) y = C (x) e A(x) + C(x) e A(x) A (x) = e A(x) (C (x) + C(x)a(x)). (24) (9) e A(x) (C (x) + C(x)a(x)) = a(x)c(x)e A(x) + b(x). e A(x) C (x) = b(x) C (x) = e A(x) b(x) C (x) C(x) : C(x) = C() + C(x) (24) ( y = C(x)e A(x) = C() + C (t) dt = C() + e A(t) b(t) dt. ) e A(t) b(t) dt e A(x) = C()e A(x) + e A(x) e A(t) b(t) dt. y() = y A(x) x =, y = y A(x) = a(t) dt y = C()e A() + e A() = C()e + = C(). C() = y 7

18 3.4 ( ) y = a(x)y + b(x) A(x) a(x) y = Ce A(x) + e A(x) e A(t) b(t) dt (C ) y() = y y = y e A(x) + e A(x) e A(t) b(t) dt, A(x) = a(t) dt. 3.5 ( ) a(x) a (25) (26) y = ay + b(x), y() = y ( ) b(x) (25) y = y = Ce ax y = ay (C ) y = C(x)e ax y = C (x) e ax + C(x) e ax a = e ax (C (x) + ac(x)). ( C() + e ax (C (x) + ac(x)) = a (C(x)e ax ) + b(x). e ax C (x) = b(x) C (x) = e ax b(x). x = y = y C(x) = C() + e at b(t) dt. ) e at b(t) dt e ax = C()e ax + e ax e at b(t) dt. y = C(). (27) y = y e ax + e ax e at b(t) dt. 8

19 5. (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 + ( + x)y = e x 2. () y + y =, y() = (2) y = x y, y() = (3) y + a y =, y() = b x (4) y ay = sin x, y() = (5) y + xy =, y() = (6) y + y tan x =, y() = 2 2π 4, ( ) 6 ( ) 4. ( y y = f x) x y x u = y y = xu u = u(x) x y = (xu ) = u + xu u + xu = f(u) u = f(u) u x u a ( a f(a) = ) ( ) du Cx = exp f(u) u (exp(t) e t ) u = y x y 5 [4] p.94, 2 6 ( ) 9

20 4. y = y2 + 2xy f(u) = u 2 + 2u a 2 + 2a = x 2 a(a + 2) = u u 2. y, y = 2x y = Cx2 Cx ( Cx = exp ) du u 2 + u = u u + = u + (C ) 4.2 (Bernoulli) n, p(x), q(x) y + p(x)y = q(x)y n n = n = n, y n = u u = ( n)y n y u u + ( n)p(x)u = ( n)q(x) 4.3 (Riccati) y = p (x)y 2 + p (x)y + p 2 (x). y y = y + u u u + (2p (x)y + p (x))u = p (x) 4.4. a, b, c b y = f(ax + by + c) u = ax + by + c u = bf(u) + a 2. a, b, c, a, b, c ( ) ax + by + c y = f a x + by + c ab a b ax+by+c =, a x+b y+c = ( ) (x, y ) x = X + x, y = Y + y ) dy dx = f ( ax + by a X + b Y ab a b = (iii) 2

21 7. () y = + 2x y (2) y = 2 y x (3) y = 4 + y2 x (4) 2 2(y x)y x 2y = (5) (2x 3y)y = x 2y (6) xy = y x 2 + y 2 (x > ) (7) x 2 y = 2(x y) 2 (8) xy 2 y = x 3 + y 3 (9) xy = y + x cos ( 2 y x) () y = cosec ( + y x ) + y x () y = 2 ( x y + y x ) (2) y = x2 + 2xy y 2 x 2 2xy y 2 (3) y = 2. () ( + x 2 )y + 4xy = 8x y (2) ( + x 2 )y 2xy + xy 2 cos x = (3) y + 2xy = 2xy 3 (4) y + xy = e x2 y 3 (5) y + y 3 e x2 xy = (6) xy + 2y = 2xy 4/3 (7) xy + 2y = y log x (8) x 2 y 2xy = y 2 cos x (9) (x )y + 2y = (x 2 )y (x > ) x + y + x y x + y x y 3. ( ) () y = xy 2 (2x )y + x (2) y = y 2 + x y 2 x 2 (3) y = y 2 3xy + 2x 2 + (4) y = y 2 4xy + 4 (5) y = y 2 2x 2 y + x 4 + 2x (6) y = y 2 + y 2x x (7) y = y 2 + 3y + 2 (8) y = y 2 2y + 4. () y = (x y) 2 (2) y = cos(x + y) (3) y = sec(x + y) (4) (x + y + )y = (5) (x + y + 2)y = x + y (6) x + y + y = x + y 5. () (x 2y )y = 2x 3y + 3 (2) (x y)y = x + y () 5. p, q I f (28) y + py + qy = f(x) 2 f(x) (29) y + py + qy = (77) m d2 x = kx (m, k ) dt2 d 2 x dt + dx 2 dt + k m x = 7 [4] p , 9,,, 2 2

22 (78) m d2 x dt 2 dx = kx γ dt (γ ) d 2 x dt + γ 2 m dx dt + k m x = (78) ( ) E(t), R, ( L), ( C) I(t) RI(t) + L di dt (t) + t I(s) dx = E(t) C t d 2 I dt + L di 2 R dt + RC I = E (t) R (77) (78) 5.2, y + py + qy = 2 λ 2 + pλ + q = (charasteristic equation), (characteristic root) 5. y 3y + 2y = λ 2 3λ + 2 = λ =, α λ 2 + pλ + q = y = e αx (78) y = e αx y = αe αx, y = α 2 e αx y + py + qy = (α 2 + pα + q)e αx = e αx =. 5.3 ( (principle of superposition)) y, y 2 (78) (3) y = Ay + By 2 (A, B ) (78) 22

23 y = Ay + By 2 y = Ay + By 2, y = Ay + By 2 y + py + qy = A(y + py + qy) + B(y + py + qy) = A + B =. λ 2 + pλ + q = 2 α, β y = Ae αx + Be βx (A, B ) (78) 5.4 y = Ae x + Be 2x (A, B ) y 3y + 2y = α β (78) (3) α = β ( 2 () ) λ 2 +pλ+ q = 2 α, β (3) y = Ae αx + Be βx (A, B ) (78) (a) A, B (79) y (78) (b) (78) A, B y = Ae αx + Be βx (a) (b) [ e αx, e βx ] y := y βy α β, y y + py + qy = y 2 := y αy β α y αy = y + y 2 = α β [(y βy) (y αy)] = (α β)y = y. α β = α β [(y βy ) α(y βy)] = α β [y (α + β)y + αβy] α β (y + py + qy) =. C y = C e αx C 2 y 2 = C 2 e βx y = y + y 2 = C e αx + C 2 e βx. 23

24 [e αx, e βx ] C, C 2 (32) C e αx + C 2 e βx = (32) α C αe αx + C 2 βe βx =. C 2 (β α)e βx =. α β C 2 =. (32) C e αx =. C =. C = C 2 = e αx, e βx 5.6 y 3y + 2y = λ 2 3λ + 2 = λ =, 2. y = Ae x + Be 2x (A, B ) λ 2 + pλ + q = α y = xe αx (78) y = xe αx y = e αx + αxe αx, y = αe αx + αe αx + α 2 xe αx = α 2 xe αx + 2αe αx y + py + qy = (α 2 x + 2α + pαx + p + qx)e αx = [ (α 2 + pα + q)x + (p + 2α) ] e αx α λ 2 + pλ + q = α 2 + pα + q =, α = p + 2 = p + 2 = p 2 y + py + qy = [ x + ] e αx =. p + 2α = 5.8 ( 2 (2) ) λ 2 + pλ + q = α (33) y = Ae αx + Bxe αx (A, B ) (78) (a) A, B (8) y (78) (b) (78) A, B y = Ae αx + Bxe αx (a) (b) 24

25 [ e αx, xe αx ] y y + py + qy = y = e αx u y = αe αx + e αx u = e αx (u + αu), y = α 2 e αx + αe αx u + e αx u = e αx (u + αu + α 2 u) y + py + qy = y 2αy + α 2 y = e αx (u + αu + α 2 u) 2αe αx (u + αu) + α 2 e αx u = e αx u. y + py + qy = e αx u =. u =. C, C 2 u = C + C 2 x. y = e αx u = C e αx + C 2 xe αx. [e αx, xe αx ] C, C 2 (34) C e αx + C 2 xe αx = C αe αx + C 2 e αx + C 2 αxe αx =. (34) α C 2 =. (34) C 2 e αx =. C e αx =. C =. C = C 2 = e αx, xe αx 5.9 y 2y + y = λ 2 2λ + = ( ). y = Ae x + Bxe x (A, B ). 5.5 (35) y + y = λ 2 + = λ = ±i (i ) H. (36) y = Ae ix + Be ix (A, B ) e ix, e ix e x d dx eλx = λe λx (λ ) (36) (35) 25

26 : z = x + iy (x, y, i ) e z = e x+iy := e x (cos y + i sin y) e z+w = e z e w Euler (37) e iy = cos y + i sin y y = π e iπ = cos π + i sin π = + i = e iπ + = (37) y y (38) e iy = cos( y) + i sin( y) = cos y i sin y (37) cos y = ( e iy + e iy), sin y = ( e iy e iy) 2 2i λ z d ( ) e λz = λe λz. dz e z = n= z n n! e x+iy = e x. p, q λ 2 + pλ + q = λ = a ± ib (a, b R; b ) Ae (a+ib)x + Be (a ib)x = Ae ax (cos bx + i sin bx) + Be ax (cos bx i sin bx) = (A + B)e ax cos bx + i(a B)e ax sin bx. C = A + B, C 2 = i(a B) (39) y = C e ax cos bx + C 2 e ax sin bx. 26

27 A, B C, C 2 8 C, C 2 (39) 5. y + y + = λ 2 + λ + = λ = ± 3i. 2 3x 3x y = Ae x/2 cos + Be x/2 sin (A, B ) y + py + qy = (p, q ) (i) λ 2 + pλ + q = 2 α, β y = Ae αx + Be βx (A, B ). (ii) λ 2 + pλ + q = α y = Ae αx + Bxe αx (A, B ). (iii) p, q λ 2 + pλ + q = a + ib (a, b R; b ) y = C e ax cos bx + C 2 e ax sin bx (C, C 2 ). 2 φ, φ 2 y = C φ (x) + C 2 φ 2 (x) (C, C 2 ) φ, φ 2 (fundamental solution). 2 α, β e αx, e βx 2. α e αx, xe αx 3. a ± ib e ax cos bx, e ax sin bx 9. () y 6y + 8y = (2) y 3y + 2y = (3) y a 2 y = (4) y + ay + k 2 y = (5) y + 2y + y = (6) y 6y + 9y = (7) y 4y + 5y = (8) y + 2y + 5y = ( ) ( ) ( ) 8 C A = C 2 i i B 9 [4] p.3 27

28 6 2 (2) (4) y + py + qy = (4) y + py + qy = f(x) ( ) ( ) 6. L[y] (42) L[y] := y + py + qy (4), (4) L[y] =, L[y] = f(x) 6. (L ) p, q x y 2 L[y] = y + py + qy (i), (ii) (i) 2 y, z (43) L[y + z] = L[y] + L[z]. (ii) 2 y k (44) L[ky] = kl[y]. (i) L[y + z] = (y + z) + p(y + z) + q(y + z) = y + z + p(y + z ) + q(y + z) = (y + py + qy) + (z + pz + = L[y] + L[z]. (ii) (43), (44) L (, linear) 5.3 ( ) y, y 2 (4) y = Ay + By 2 (4) y, y 2 (4) L[y ] =, L[y 2 ] = L[y] = L[Ay + By 2 ] = AL[y ] + BL[y 2 ] = A + B = y (4) 6. 28

29 6.2 ( ) j =, 2 y j y + py + qy = f j (x) y = y + y 2 y + py + qy = f (x) + f 2 (x) L[y] = L[y + y 2 ] = L[y ] + L[y 2 ] = f (x) + f 2 (x). 6.3 y = 4 ex y 6y + 9y = e x y 2 = 9 x y 2 6y 2 + 9y 2 = x y = y + y 2 = 4 ex + 9 x y 6y + 9y = e x + x ( ) u (4) (i), (ii) (i) z (4) y = u + z y (4) (ii) y (4) z = y u z (4) u z y = u + z z (4) y (4). (i) L[y] = L[u + z] = L[u] + L[z] = f(x) + = f(x). (ii) L[z] = L[y u] = L[y] L[u] = f(x) f(x) =. = + X = (4), X f = (4) 29

30 X f = {u + z; z X }. 6.5 (45) y 3y + 2y = + x u = 2 x ( ) z 3z + 2z = (45) z = Ae x + Be 2x (A, B ) y = u + z = Ae x + Be 2x + 2 x (A, B ). 6.3 ( ) f ( ) ( ) (a) f(x) = (x n ) e αx α m (m ) u(x) = (n ) x m e αx L[u] = { } (b) f(x) = (x n ) e ax cos bx ( a, b R) a + ib m sin bx (m ) u(x) = (n ) x m e ax (A cos bx + B sin bx) 6.6 L[y] = y 5y + 6y () L[y] = 6x 2 + 2x 2 (2) L[y] = e 2x (3) L[y] = sin x λ 2 5λ + 6 = λ = 2, 3. L[z] = z = Ae 2x + Be 3x (A, B ) () ( α =, m =, n = 2) u = ax 2 + bx + c u = 2ax + b, u = 2a 3

31 L[u] = 2a 5(2ax + b) + 6(ax 2 + bx + c) = 6ax 2 + (6b a)x + (2a 5b + 6c). 6x 2 + 2x 2 6a = 6, 6b a = 2, 2a 5b + 6c = 2 a =, b = 2, c =. u = x 2 + 2x +. y = z + u = Ae 2x + Be 3x + x 2 + 2x +. (2) 2 ( ) ( α = 2, m =, n = ) u = axe 2x u = ae 2x + 2axe 2x, u = 3ae 2x + 4axe 2x L[u] = (3ae 2x + 4axe 2x ) 5(ae ax + 2axe 2x ) + 6axe 2x = 2ae 2x. e 2x 2a =. a = /2, u = xe 2x /2. y = z + u = Ae 2x + Be 3x xe2x 2. (3) ±i ( a =, b =, m =, n = ) u = a cos x + b sin x (a, b ) u = a sin x + b cos x, u = a cos x b sin x L[u] = ( a cos x b sin x) 5( a sin x + b cos x) + 6(a cos x + b sin x) = (5a 5b) cos x + (5b + 5a) sin x. sin x 5a 5b =, 5a + 5b =. a = b = /. u = (cos x + sin x). y = z + u = Ae 2x + Be 3x + (cos x + sin x). 2. () y 6y + 8y = e x (2) y 6y + 8y = 3e 2x (3) y 3y + 2y = sin x (4) y 3y + 2y = e x (5) y a 2 y = xe ax (6) y + a 2 y = x 2 (7) y + 2y + y = e x (8) y + 2y + y = x 2 (9) y 6y + 9y = x + e x () y 6y + 9y = cos x () y 2y = + x 2 [4] pp

32 7 2 (3) f(x) (46) y + py + qy = f(x) () Laplace, (2) 2 Green 7. (Green ) 2 λ 2 + pλ + q = 2 α, β e αx e βx (α β ) (47) G(x) = α β xe αx (α = β ), (48) u(x) = G(x y)f(y) dy u + pu + qu = f(x), u() = u () = u (46) G (46) Green 7.2 ( ) 7.3 [, ) f, g f g (f g)(x) = f(x y)g(y) dy (x [, )) f g 7.4 e n (x) (n =, 2, ) e (x) =, e k+ = e e k (k =, 2, ) (49) e n (x) = xn (n )! (49) n = k e k+ (x) = (e e k )(x) = e (x y)e k (y) dy = y k (k )! dy = (49) n [ y k k! ] x = xk k! 2 32

33 (66) u u = G f 7.5 ( ) () (c f + c 2 f 2 ) g = c (f g) + c 2 (f 2 g). (2) f g = g f. (3) (f g) h = f (g h). (4) f g f g. () (2), (3) (4) ( [34] ) y ay = f(x), y() = y = e a(x y) f(y) dy y = (e ax f)(x) A(x) = e αx, B(x) = e βx u v = u βu u + pu + qu = f(x), u() = u () = v αv = (v βv ) α(v β) = v (α + β)v + αβv = v + pv + qv = f(x), v() = u () βu() = β = u = B v. v(x) = (A f)(x). u βu = v(x), u() = u = B v = B (A f) = (B A) f. G = B A u = G f G α β G(x) = α = β G(x) = B(x y)a(y) dy = = e βx [ e (α β)y α β ] x = eαx e βx α β. B(x y)a(y) dy = 33 e β(x y) e αy dy = e βx e (α β)y dy e α(x y) e αy dy = e αx dy = xe αx.

34 . f g = g f, (f g) h = f (g h) ( : ) 2. G y + py + qy = (p, q ) Green G + pg + qg =, G() =, G () = Green 8 ( ) ( ) 8. (f ) f(x, y) y = f(x, y), y(x ) = y δ (x δ, x + δ) u u (x) = f(x, u(x)) (x (x δ, x + δ)), u(x ) = y ( ) 34

35 8.3 δ 8.3 dy dx = y2 y() = a (a ) y = /a x (x (, /a)). lim y =, x /a ( /a) x /a (f ) 23 y Lipschitz L f(x, y ) f(x, y 2 ) L y y 2. f(x, y) y (f(x, y) = A(x)y + b(x) ) ( ) ( A(x) ) ( ) 8.4 (5) y = y 2/3, y() = y y = { (x C ) 3 (x < C ) (x C ), y = { (x < C 2 ) (x C 2 ) 3 (x C 2 ), (x C ) 3 (x < C ) y = (C x C 2 ) (x C 2 ) 3 (x > C 2 ) ( C C 2 ) 23 Rudolf Otto Sigismund Lipschitz (832 93, Königsberg Bonn ) 35

36 8.4 ( ) f(x, y) y f(x, y ) f(x, y 2 ) L y y 2 (L ) dy dx = f(x, y) (x [a, b]), y(a) = y y = φ (x) (x [a, b ]), y = φ 2 (x) (x [a, b 2 ]) φ (x) = φ 2 (x) (x [a, b ], b := min{b, b 2 }) ψ(x) := φ (x) φ 2 (x) (x [a, b ]) φ j (x) = y + a f(s, φ j (s)) ds (j =, 2) ψ(x) = a [f(s, φ (s)) f(s, φ 2 (s))] ds ψ(x) a f(s, φ (s)) f(s, φ 2 (s)) ds L φ (s) φ 2 (s) ds = L a a ψ(s) ds. M := max ψ(x) x [a,b ] ψ(x) LM(x a), ψ(x) L LM(s a) ds = L 2 (x a)2 M, a 2 ψ(x) L L 2 (s a)2 M ds = L 3 (x a)3 M, 2 3! a [L(x a)]n ψ(x) M M [L(b a)] n (n N). n! n! x ψ(x) φ (x) = φ 2 (x) (x [a, b ]). Lipschitz ( L ) f C 36

37 8.5 (C ) f C dy dx = f(x, y) (x [a, b]), y(a) = y y = φ (x) (x [a, b ]), y = φ 2 (x) (x [a, b 2 ]) φ (x) = φ 2 (x) (x [a, b ], b := min{b, b 2 }) f(x, y) = y 2/3 y = f Lipschitz y = f(x, y), f(x ) = y () f (x, y ) (2) f C (3) f C y f(x, y ) f(x, y 2 ) L y y 2 (L ) 37

38 . () 2, y = x 2 yy + 3 x (2), y = ± y 2 + log( + x 2 ) (3) 3, y = (y ) 2 y = 2x x 2 y 6 x y (5) 4, 2 y(4) = 2 2 y(3) 3 y 3 ex (6) 3, y = x y, y = ± k( + (y ) 2 ) (8) 2, y = x 2 y 4y (4) 2, y ( x ) y (7) () y = + C x2 (2) y = x 4 + C log x + C 2 (3) x 2 + a 2 y 2 = C (4) y = C x + C 2 (5) y 2 + 2xy = C x ( ) 2 2 (6) y = sin π tan x + C (7) a 2 y 2 = (C ± x) 2 (8) y = a sin(c ± x) (9) y = C sin(c 2 ± x) ( ) () y = x C + f(x) dx 2. () y = 2x2 Cx 2 (2) y = (x + C)3 (3) y = 4 (5) y 2 = 5x2 Cx 5 2 (6) y = Cx x + (7) y = ( x 2 + ) (4) y = x 2 + ax + C (8) y = cot (log cos x + C) ( ) (9) y log y = x + log C x () sin x cos y = C () y = 3 log + x3 + C x Cx (2) y = b(c + e2abx ) C e 2abx (3) y = tan (tan x + C) (4) y = tan (x cos x + C) (5) y = C x 2 (6) 2 y2 + log y = 2 x2 + log C x (7) y log y + = 2 x2 + C (8) y = C + x2 C x (9) y = 2 2 log( e2x + C) (2) y = log(c e x ) (2) y = Ce x (C > ) y = Ce x (C < ) (22) y 2 = x 2 + C (23) y = ( x + C) 2 (24) y 3/2 = x 3//2 + C x (25) y = Cx + (26) y = 2x2 Cx 2 + (27) y2 = ( + x 2 + C) 2. () y 2 = C(x 2 + ) 2 2 (2) y = exp( exp(2x) + C) (3) y = tan log C x (4) y = cot (tan x + C) (5) log y + 4 y2 = x 2 + C (6) y = C sin x (7) y2 = C(x 2 + 2) 4 (8) y = log(e x + C) (9) y 2 = C(x 2 + ) 3. () y = Ce ax (2) y = Ce ax + b a, a (3) y = (C + x) cosec x (4) y = + Ce x2 2 (5) y = C sec x 2 cos x (6) y = + xe Cex2 x2 (7) y = C x + 2 (x log x 2 x) (8) y = Ce ax + ( ) a + b ebx x 2 (9) y = C x a () y = C exp () y = C 2 x + 2 x 4 x3 (2) y = C + 2x + x3 x (3) y = Cx + 2 x(x cos x sin x) (4) y = (C + cos x)e sin x (5) y = Cx x 2 log( x2 ) 38

39 (6) y = Ce ax + a (a sin x + cos x) (7) y = + x 2 (C + tan x) (8) y = Ce x x3 /3 + e x3 /3 2 (9) y = Ce ax + b a x2 + ( c 2b ) x + ( d c a a a a + 2b ) (2) y = C a 2 x e x + 2x e3 2. () y = e x + (2) y = e x + x (3) y = bx a (4) y = + a 2 (e ax a sin x cos x) (5) y = e x2 /2 (6) y = 2 cos x 2π 4. () (x + y)(y 2x) 2 = C (2) y = C ( ) x + x (3) y = x 2 (4) x 2 + 4xy 2y 2 = C log x + C (5) (x y)(x 3y) = C (6) y + ( ) x 3 2C x 2 + y 2 = C (7) y = x (8) y 3 = 3x 3 log Cx (9) 2x 3 C y = x tan (log x + C) () y = x(π cos log Cx) () x 2 y 2 = Cx (2) x 2 + y 2 = C(2x + y) (3) x + y + x y = C 2. () y = ( ) 2 C x (2) y = x 2 + x sin x + cos x + C (3) y2 = + Ce 2x2 (4) (C 2x)y 2 = e x2 (5) (C + 2x)y 2 = e x2 (6) y /3 = 2x + Cx 2/3 (7) 2 y = log x + C x (8) y(c sin x) = x2 (9) [ y = x x2 log(x + x 6(x ) 2 ) + C ] 2 2 ( ) 3. (), (y )( x+ce x ) = (2) 2, (y 2)( 3x+Ce 3x ) = 9x (3) x, (x y) e x 2 /2 dx + C = ( e x2 /2 (4) 4x, (y 4x) C ) e 2x2 dx = e 2x2 (5) x 2, (y x 2 )(C x) = (6) x, (y x) ( 2 ) + Ce (4/3) x 3 x (7), (y + )( Ce x ) + = (8), (y )(C x) = = 4. () x y = coth(x + C) (2) tan ( ) x+y 2 = x + C (3) sin(x + y) = x + C (4) x + y + 2 = Ce y (5) x + y + = Ce x y (6) (x + y + ) 2 (x + y) (x + y) 2 + cosh (x + y) = 4x + C 5. () C(y x 4) 2 = e (x+9)/(y x 4) (2) x = Ce t cos t 2, y = Cet sin t 2 5. () y = C e 2x + C 2 e 4x (2) y = C e x + C 2 e 2x (3) y = C e ax + C 2 e ax (a ), y = C e 2x + C 2 x (a = ) (4) y = C e ( a/2+ a 2 /4 k a 2 )x + C 2 e ( a/2 2 /4 k 2 )x ( a 2k ), y = C e (a/2)x + C 2 e (a/2)x ( a = 2k ) (5) y = C e x + xc 2 e x (6) y = C e 3x + C 2 xe 3x (7) y = C e 2x cos x + C 2 e 2x sin x (8) y = C e x cos 2x + C 2 e x sin 2x 6. () y = 3 ex + C e 2x + C 2 e 4x (2) ( x 2 (3 cos x + sin x) + C e x + C 2 e 2x (3) 4a x ) e ax + C 4a 2 e ax + ( ) x 2 C 2 e ax (4) y = 2 + C + C 2 x e x (5) y = x 2 4x (C + C 2 x)e x 39

40 (6) y = x ex 4 + (C + C 2 x) e 3x (7) y = 5 (4 cos x 3 sin x) + (C + C 2 x)e 3x (8) y = 4 (x2 + 3x) + C + C 2 e 2x 4

41 A A. ( ) A.2 ( ) B 2 p, q f(x) (5) y + py + qy = f(x) Green B. ( ) (a) 24 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 (x)y + c 2 (x)y 2 y = [c (x)y + c 2(x)y 2 ] + [c (x)y + c 2 (x)y 2]. y (5) c (x)y + c 2(x)y 2 = 24 ( ) 4

42 (c, c 2 ) y = c (x)y + c 2 (x)y 2, y = [c (x)y + c 2(x)y 2] + [c (x)y + c 2 (x)y 2] L[y] = y + py + qy = c (x)l[y ] + c 2 (x)l[y 2 ] + [c (x)y + c 2(x)y 2] = c (x)y + c 2(x)y 2. L[y] = f(x) (52) c (x)y + c 2(x)y 2 = f(x) (6), (6) c (x), c 2(x) c (x), c 2 (x) y = c (x)y + c 2 (x)y 2 B. (2 ) L[y] = y 6y +8y = e x. L[z] = z = Ae 2x + Be 4x. y = c (x)e 2x + c 2 (x)e 4x y = ( c (x)e 2x + c 2(x)e 4x) + ( c (x)2e 2x + c 2 (x)4e 4x) (53) c (x)e 2x + c 2(x)e 4x = y = 2c (x)e 2x + 4c 2 (x)e 4x. y = ( 2c (x)e 2x + 4c 2(x)e 4x) + ( 4c (x)e 2x + 6c 2 (x)e 4x). L[y] = ( 2c (x)e 2x + 4c 2(x)e 4x) + ( 4c (x)e 2x + 6c 2 (x)e 4x) 6 ( 2c (x)e 2x + 4c 2 (x)e 4x) + ( c (x)e 2x + c 2 (x)e 4x) = 2c (x)e 2x + 4c 2(x)e 4x. e x (54) 2c (x)e 2x + 4c 2(x)e 4x = e x. (62), (63) ( e 2x e 4x c, c 2 ( c (x) c 2(x) ) 2e 2x = ( 4e 4x e 2x 2e 2x ) ( e 4x 4e 4x 42 c (x) c 2(x) ) ( ) e x = ) ( = e x ). 2 e x 2 e 3x.

43 c, c 2 ( c (x) c 2 (x) ) = 2 e x 6 e 3x u = c (x)e 2x + c 2 (x)e 4x = 2 e x e 2x 6 e 3x e 4x = 3 ex L[y] = e x y = Ae 2x + Be 4x + 3 ex. B.2 Oliver Heaviside 25 (85 925, ) ( ) p p pf(x) = d dx f(x), p f(x) = f(t) dt. Laplace (T.Bromwich ) 2. (Jan Mikusiński) [3], [35] Laplace ( ) [35] ([3] ) ( ) [8] Laplace Laplace ( ) 25 Maxwell ( Maxwell Heaviside 43

44 B.3 Laplace Laplace B.4 α, β (D α) ((D β)y) = f(x) (D a)y = f(x) y ay = f(x) y = Ce a(x x ) + x e a(x t) f(t) dt (C ) B.2 α, β λ 2 + pλ + q = f I x I ( t ) (55) u(x) = e α(t s) f(s) ds dt x x e β(x t) u + pu + qu = f(x), u(x ) = u (x ) = x = Green B.5 Green n 7 n α,, α n y (n) + a y (n ) + + a n y + a n y = f(x) G(x) = e α x e α 2x e αnx, u(x) = G f(x) u u (n) + a u (n ) + + a n u + a n u = f(x), u() = u () = = u (n ) () = 44

45 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 Laplace G = s α s α n n j= A j s α j, A j = j k (α k α j ) G(x) = n A j e α jx j= G : G (n) + a G (n ) + + a n G + a n G =, G() = G () = = G (n 2) () =, G (n ) () =. α = = α n = α G(x) = xn e αx (n )! C. 2. IV 3. D 23 IV D. 23 IV D.. 2 IV IV 45

46 D..2 OK D.2 IV ( ) y = g (g ) y = f(x) y = f(x) y = gx + C (C ), y = g 2 x2 + Cx + C (C ). y = f(x)g(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 by)y ( ), (56) y = a(x)y ( ) (57) y = a(x)y + b(x) ( ) y = ay ( ) y = a(x)y y = a(x)y + b(x) 2 D. ( ) L[y] = y a(x)y L[y + z] = L[y] + L[z], L[ky] = kl[y] L (57), (56) ( (58), (59) 26 ) y 26 L[y] = y + py + qy L[y + z] = L[y] + L[z], L[ky] = kl[y] 46

47 ( ) 2 (58) y + py + qy = (59) y + py + qy = f(x) ( ) (58) 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 (59) u y = u + z (z L[z] = ) ( ) (a) (b) (c) (d) Laplace (e) y = g ( ) y = ay ( ), y = ω 2 y ( ) 47

48 2 L[y] L[y + z] = L[y] + L[z] 2 or E 2 E. ( ) (a) 27 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 (x)y + c 2 (x)y 2 y = [c (x)y + c 2(x)y 2 ] + [c (x)y + c 2 (x)y 2]. y (6) c (x)y + c 2(x)y 2 = (c, c 2 ) y = c (x)y + c 2 (x)y 2, y = [c (x)y + c 2(x)y 2] + [c (x)y + c 2 (x)y 2] 27 ( ) 48

49 L[y] = y + py + qy = c (x)l[y ] + c 2 (x)l[y 2 ] + [c (x)y + c 2(x)y 2] = c (x)y + c 2(x)y 2. L[y] = f(x) (6) c (x)y + c 2(x)y 2 = f(x) (6), (6) c (x), c 2(x) c (x), c 2 (x) y = c (x)y + c 2 (x)y 2 E. (2 ) L[y] = y 6y +8y = e x. L[z] = z = Ae 2x + Be 4x. y = c (x)e 2x + c 2 (x)e 4x y = ( c (x)e 2x + c 2(x)e 4x) + ( c (x)2e 2x + c 2 (x)4e 4x) (62) c (x)e 2x + c 2(x)e 4x = y = 2c (x)e 2x + 4c 2 (x)e 4x. y = ( 2c (x)e 2x + 4c 2(x)e 4x) + ( 4c (x)e 2x + 6c 2 (x)e 4x). L[y] = ( 2c (x)e 2x + 4c 2(x)e 4x) + ( 4c (x)e 2x + 6c 2 (x)e 4x) 6 ( 2c (x)e 2x + 4c 2 (x)e 4x) + ( c (x)e 2x + c 2 (x)e 4x) = 2c (x)e 2x + 4c 2(x)e 4x. e x (63) 2c (x)e 2x + 4c 2(x)e 4x = e x. (62), (63) ( e 2x e 4x 2e 2x 4e 4x ) ( c (x) c 2(x) ) = ( e x ). c, c 2 ( c (x) c 2(x) ) = ( e 2x 2e 2x e 4x 4e 4x ) ( e x ) = 2 e x 2 e 3x. 49

50 c, c 2 ( c (x) c 2 (x) ) = 2 e x 6 e 3x u = c (x)e 2x + c 2 (x)e 4x = 2 e x e 2x 6 e 3x e 4x = 3 ex L[y] = e x y = Ae 2x + Be 4x + 3 ex. E.2 Oliver Heaviside 28 (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 [35] [3] 28 Maxwell ( Maxwell Heaviside html ) 5

51 Laplace [35] ([3] ) ( ) [8] Laplace Laplace ( ) 29 E.3 Laplace Fourier Laplace E.4 L (n ) 2 y + py + qy = ( ) 2 ( ) d d y + p y + qy = dx dx D = d dx D 2 y + pdy + qy = (D 2 + pd + q)y = 29 5

52 n F (λ) = a j λ j j= F (D)y := n j= a j d j y dx j D 2 + pd + q { (F (D) + G(D))y = F (D)y + G(D)y, (F (D) G(D))y = F (D)(G(D)y) λ 2 + pλ + q = (λ α)(λ β) L[y] = (D 2 + pd + q)y = ((D α)(d β))y = (D α)((d β)y) L[y] = f (D α)(d β)y = f v := (D β)y (D α)v = f. v(x ) = C (64) v(x) = Ce α(x x ) + x e α(x t) f(t) dt (C ) 3 v y : (D β)y = v (65) y(x) = C e β(x x ) + (64) (65) y(x) = C e β(x x ) + v(t) = Ce α(t x ) + x e β(x t) t x e β(x t) v(t) dt. x e α(t s) f(s) ds ( t ) Ce α(t x) + e α(t s) f(s) ds dt. x 3 x x = 52

53 C = C = ( t ) u(x) = e α(t s) f(s) ds dt. x x e β(x t) 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 ) = E.2 α, β λ 2 + pλ + q = f I x I ( t ) (66) u(x) = e α(t s) f(s) ds dt x x e β(x t) y + py + qy = f(x) ( u(x ) = u (x ) = ) x = u = e αx e βx f(x) (i) α β (ii) α = β u(x) = G f(x) := G(x) = G(x) := e αx e βx G(x) = G(x y)f(y) dy e α(x y) e βy dy = eαx e βx α β. e α(x y) e αy dy = xe αx. 53

54 E.3 ( ) α, β λ 2 + pλ + q = f I e αx e βx (α β) (67) u(x) := G(x y)f(y) dy, G(x) := α β xe αx (α = β) u + pu + qu = f(x), u() = u () = G Green Green (?) Green G.5. ( E.2) Mathematica E.2 ( ) Mathematica 3 I.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 = e 4(x t) ( t ) e 2(t s) e s ds dt = 2 e2x + 6 e4x + 3 ex. (2) special[2,,sin[s]] y = e 2(x t) ( t ) e (t s) sin s ds dt = 2 ex + 5 e2x + 3 cos x + sin x. (3) special[-a,a,exp[a s]] y = e a(x t) ( t (4) special[-,-,exp[-s]] 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 ) e (t s) e s ds dt = 2 x2 e x. 3 54

55 (5) special[-,-,s^2] y = e (x t) ( t (6) special[3,3,(s+exp[s])] y = (7) special[3,3,cos[s]] y = (8) special[2,,+s] e 3(x t) ( t e 3(x t) ( t y = ) 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. ( ) f g(x) = f g f g f(x t)g(t) dt 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) E.5 Green n n y (n) + a y (n ) + + a n y + a n y = f(x) α,, α n G(x) = e α x e α 2x e αnx, u(x) = G f(x) 55

56 u 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, s α j j= A j = j k (α k α j ) 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 ) () =. 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 () = F Laplace Laplace L.Euler P.S. de Laplace ( ) (Euler ) Heaviside ( ) Heaviside ( ) Laplace [32] [9], [28] [33] Laplace ( ) Laplace Laplace ( ) 56

57 F. F. (Laplace ) f C([, ); C) L[f](s) := e sx f(x) dx L[f] f Laplace F.2 ( ) L[f + g](s) = L[f](s) + L[g](s). F.3 ( 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] = s s 2 + ω 2, L [sin ωx] = ω s 2 + ω 2. (5) ( Laplace ) L [e ax xα Γ(α) (s a)x = y [ ] L e ax xα (s) = Γ(α) = L [cosh ωx] = ] (s) = s s 2 ω 2, L [sinh ωx] = ω s 2 ω 2. e sx ax xα e Γ(α) dx = Γ(α) e (a s)x x α dx ( ) α y e y dy Γ(α) s a s a = (s a) α Γ(α) (s a) α Γ(α) Γ(α) = (s a). α e y y α dy () α = ( ) 57

58 L [e ax ] (s) = e sx e ax dx = (2) e (a s)x dx = [ ] e (a s)x a s = ( ) = a s s a. L[](s) = e sx dx = s [ e sx ] = s ( ) = s. (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 sx = y 32 L [x n ] (s) = = ( y ) n 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) = 2 e sx cos ωx dx ( e sx cos ωx ) ( e sx sin ωx ) = se sx cos ωx + ( ω)e 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 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 ) 32 Γ 58

59 (5) L[cosh ωx](s) = L [ (e ωx + e ωx )/2 ] = 2 = s s 2 ω, 2 L[sinh ωx](s) = L [ (e ωx e ωx )/2 ] = 2 = ω s 2 ω. 2 F.4 (δ Laplace ) ( L [e ωx ] (s) + L [ e ωx] (s) ) = ( 2 s ω + ) s + ω ( L [e ωx ] (s) L [ e ωx] (s) ) = ( 2 s ω ) s + ω L[δ](s) =. Laplace F.5 (Laplace ()) L [ f (n)] n (s) = s n L[f](s) s j f (n j) () n = L[f ](s) = n OK j= = s n L[f](s) s n f() s n 2 f () sf (n ) () f n () = + s e sx f (x) dx = [ e sx f(x) ] ( s)e sx f(x) dx 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) F.6 (Laplace ) [ L L[F ](s) = = s n s j f (n+ j) (). j= ] f(t) dt (s) = s L[f](s). 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 59

60 F.7 (Laplace (2)) d ds L[f](s) = d ds d L[f](s) = L [ xf(x)] (s). ds ( ) n d L[f](s) = L [( x) n f(x)] (s). ds e sx f(x) dx = F.8 ( Laplace ) F.9 F. L [e αx f(x)] (s) = L[f g](s) = L[f g](s) = L[f](s)L[g](s). = = e sx ( dy y e sy g(y) dy = L[g](s)L[f](s). [ ] L x f(x) (s) == e sx ( x)f(x) dx = L [( x)f(x)] (s). ) f(x y)g(y) dy dx e sx f(x y)g(y) dx s e st f(t) dt L[f](t)dt. 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 α)x f(x) dx = L[f](s α). (s α) n. F. ( Laplace ) f T L[f](s) = L[f](s) = e st e sx f(x) dx = 6 T n= e sy f(y) dy. (n+)t nt e sx f(x) dx.

61 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. F.2 F.2 Laplace y 5y + 6y = x + sin x + e 3x ( s 2 L[f](s) sf() f () ) (sl[f](s) f()) + 6L[f](s) = s 2 + s s 3. L[f](s) (s 2 5s + 6)L[f](s) = f () + f()(s 5) + s 2 + s s 3. 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). F. (Mathematica ) Mathematica Apart[] solution=solve[(s^2-5s+6)y==/s^2+/(s^2+)+/(s-3),y] Ly= y /. solution[[,]] Ly=Apart[Ly] Mathematica Laplace InverseLaplaceTransform[] InverseLaplaceTransform[Ly,s,x] 6

62 e2x 7 9 e3x + x 6 + xe3x + (cos x + sin x) ( ) F.3 ω R L [ e iωx] (s) = s iω = s + iω (s iω)(s + iω) = s s 2 + ω + i ω 2 s 2 + ω. 2 F.4 L[cos ωx](s) = s s 2 + ω, L[sin ωx](s) = ω 2 s 2 + ω. 2 L[f ](s) = s 2 L[f](s) sf() f () f(x) = sin ωx ω 2 L[sin ωx](s) = s 2 L[sin ωx](s) s ω. F.5 f (s 2 + ω 2 )L[sin ωx](s) = ω. L[sin ωx](s) = L[f](s) = ω s 2 + ω 2. s 2 + 2s s 2 + 2s = s(s + 2) = ( 2 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 + 2 e 2t dt = 2 ( e 2x ). F.6 n f n f(x) = L [ s n ] (x) = L [ s f (x) f n (x) = xn (n )!. L[f n ](s) = s n ] (x) = s n 2 62 [ ] L (t) dt = x n f n (t) dt.

63 F.7 f L[f](s) = ( ) d = ds s 2 + ω 2 s (s 2 + ω 2 ) 2 2s (s 2 + ω 2 ) 2 s = ( ) d (s 2 + ω 2 ) 2 2 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.8 [ ] 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 ). F.9 ) 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 xf(x) = 2(cos ωx ). f(x) = 2( cos ωx). x F.2 H Heaviside { (x ) H(x) = (x < ) L [H(x a)f(x a)] = e as L[f](s). 63

64 L [H(x a)f(x a)] = = F.2 f e sx H(x a)f(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 F.3 s C Lebesgue e s x f(x) dx 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) Laplace ( M R) ( α R) ( x R) f(x) Me αx 33 s > α Laplace F.4 Fourier, Laplace f Fourier g Fourier F[f](y) = e ix y f(x) dx, F [g](x) = e ix y g(y) dy 2π 33 Laplace 64

65 Laplace L[f](s) = e sx f(x) dx (Re s > a), F[f](y) = L[f](iy) 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 < ). [4] ( ) F.22 ( Laplace ) () P (x) n a,, a n [ ] L P (s) (x) = (s a ) (s a n ) (2) P (x) n a C [ ] P (s) L (x) = (s a) n n k= n k= P (a k )e a kx j k (a k a j ). P (n k) (a)x k e ax (n k)!(k )!. F.5 Laplace Schwartz [] Yosida [34] L 2 Laplace F.23 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 65

66 F.24 (Paley-Wiener) g H 2 () g y g(iy) L 2 (R) : lim g(x + iy) g(iy) x R 2 dy = f(t) := N 2π l.i.m. g(iy)e ity dy N N ( L 2 ). f (, ) f Laplace g Payley-Wiener Yosida [34] Laplace Schwartz [] Schwartz [9] Laplace [] [] ([36] I 4 ) F.6 ( ) Banach X C {U(t)} t Ax = lim (U(h) I)x h + h X A A D(A) X 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(t)x dt (x X, Re λ > β). s a = L [ e at] (s) c+it U(t)x = lim e λt (λ A) x dλ (c > β, t >, x D(A)) T c it ({U(t)} x X t ) A L(X) (X ) U(t) = e ta def. = U(t)x = lim n e tan x, n= t n n! An ( ) ( U(t)x = lim + t ) n n n A x, A n = A 66 ( + n A ) ( ).

67 F.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 ω sinh ωx s > ω s 2 ω 2 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 a)h(x a) f(ax) (a > ) e as φ(s) ( s ) a φ a G ( ) 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(d)y = f(x) 67

68 G. ( ) f(x) = n j= a jx j f(d)y := n a j y (j) j= (f(d) + g(d))y = f(d)y + g(d)y, (f(d)g(d)) y = f(d) (g(d)y) (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 + S)x := 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) S T (RS)T = R(ST ) L(X) C (algebra) id X : X x x X I 68

69 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 n := T} T {{ 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 L(X), f(x) = T n+m = T n T m, (T n ) m = T nm (n, m Z) n a j x j C[x] j= f(t ) L(X) f(t ) = f(t )y = n a j T j j= n a j (T j y). j= 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)}) 69

70 T L(X) R(T ) := {T x; x D(T )} T T L(X) T T S : R(T ) y x X ( T x = y x) G.2 D I R X = C (I; C) G. ( D) D L(X) Dy = y (y ) (C X m := C m (I; C), X := X D D(D) X ) G.2 ( ) f X T f y := fy ( ) T f L(X) f ( α e αx ) G.3 f(x) C[x], α C f(d)e αx = f(α)e αx. D k e αx = α k e αx (k =,, 2, ) ( ()) G.4 ( ) u X, α, β C (), (), (2) () D(e αx u) = e αx (D + α)u. () m N (2) m N () e αx (D α) m (e αx u) = D m u. 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 + α)u. 7

71 () () e αx D (e αx u) = (D + α)u. e αx αe αx u = αu e αx (D α)e αx u = Du. m [ e αx (D α)e 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 α)e αx u + (α β)u = Du + (α β) = [D + (α β)] u. (Cf. P (λi A)P = λi P AP.) () m G.3 G.3. G.5 ( ) f, g C([, ); C) f g (, convolution) f g f g(x) = f(x y)g(y) dy (x [, )). ( f + g h = f + (g h) ) G.6 ( ) ( ) () ( ) 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) [34] ( ) f, g f g() = 7

72 d dx a f(x, t) dt = f(x, x) + a df (x, t) dt dx G.7 ( ) 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 y)g(y) dy. j= (f g) (x) = f()g(x) + (f g) (x) = f()g (x) + f ()g(x) + f (x y)g(y) dy, f (x y)g(y) dy, (f g) (3) (x) = f()g (x) + f ()g (x) + f ()g(x) + f (3) (x y)g(y) dy, G.8 f C k ([, ); C), g C k ([, ); C), f() = f () = = f (k ) () = (f g) (r) () = (R =,,, k). x = G.9 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) () =. G.8 f() = f () = f (m 2) () =. G.3.2 e m,α e m (x) = xm (m )! (m =, 2, ) e m(x) = e m (x) e (l) m = e m l 72

73 G. (e m,α ) α C, m Z x m (68) e m,α (x) := (m )! eαx (m ) (m ) e m,α e m, (x) = x m /(m )!, e,α (x) = e αx G. (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 α)e m,α (x) = e m,α (x). G.2 P = (e,α, e 2,α,, e m,α ), J = G.3 u(x) = (D α)p = P J. m c j e j,α (x) j= G. c j = (D α) j u() x = (D α) l u(x) = e k,α () = { (j =, 2,, m). m c j e j l+,α (x) j=l (k = ) (k ) G.4 (D α) m u = f G.4 73

74 G.4 ((D α) m u = ) α C, m N, I R u C m (I; C) (i) (D α) m u =. m (ii) (c,, c m ) C m s.t. u(x) = 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 = (ii) e αx (D α) m e αx v = D m v = (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) j= G.3 c j = (D α) j u() I ( ) G.5 ((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), u() = u () = u (m ) () = v() = v () = v (m ) () = 74

75 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) = xm (m )! F (x) = v(x) = } {{ } F (x). m } {{ } (x) = m xm (m )! (x y) m F (y) dy = (m )! (x y) m e αy f(y) dy. (m )! u = e αx v u(x) = ( ) (x y) m e α(x y) f(y) dy. (m )! G.6 ((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). G.5 p(d)u = f p(x) C[x] p(d)u = f p(x) (69) p(x) = r (x λ j ) m j (λ j, a C; j k = λ j λ k, m j ) j= 75

76 G.7 (e k,λj p(d)y = ) p(x) (69) (7) e k,λj (x) = xk (k )! eλ jx (j =, 2,, r; k =, 2,, m j ) p(d)y = c jk y = m r j c jk e k,λj (x) j= k= p(d)y = 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(d)y = G.8 (7) {e k,αj } (7) 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) = (7) T mj ( r m j ) = T mj c jk e k,λj (x) = j= k= { e,λj (x) = e λ J x (k = m J ) (k < m J ) [ ] ( 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 =. 76

77 (7) j J (D λ j) m j ( ) G.9 ( p(d)y = ( )) p(x) (69) p(d)y =, D = d dx C (I; C) n e k,λj (x) = xk (k )! eλ jx (j =, 2,, r; k =, 2,, m j ) y = p(d)y = m r j c jk e k,λj (x) j= k= (c jk ) G.2 ( p(d)y = f ) p(x) (69) 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(d)u = f, u() = u () = u () = = u (n ) () = y = e m,λ f (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 77

78 p(d)y 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 y)f(y) dy G Green Green n = 2 e αx e βx (72) 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 α j (j =, 2,, n) e α x e αnx = n e α jx (α j α k ). j= k j (72) α β Laplace L [e α x e αnx ] (s) = n L [e αjx ] (s) = j= n j= s α j. n j= s α j = n j= A j s α j s = α l = n A j (s α k ) j= k j [ = A l (α l α k ) A l = (α l α k )]. k l k l 78

79 e α x e αnx = n A j e αjx. (Laplace Green ) Green p(s) = r (s α j ) m j j= ( Heaviside ) Green [2] 34 G.2 (Green ) n p(d) (p(x) C[x]) p(d)g(x) =, j= G() = G () = = G (n 2) () =, G (n ) () = G f C([, ); C) u := G f p(d)u = f, u() = u () = = u (n ) () = G u(x) = G(x y)f(y) dy 34 ( [7] Green ( ) Green [2], [2] ( [33]) u = G f [2] [2] ) ( Green Green ) ( ) 79

80 u (x) = G()f(x) + G (x y)f(y) dy, u (x) = G()f (x) + G ()f(x) +. u (r) (x) =.. r G (j) ()f r j (x) + j=. G (x y)f(y) dy, G (r) (x y)f(y) dy, u (n ) (x) = G()f (n 2) (x) + + G (n 2) ()f(x) + G (n ) (x y)f(y) dy, u (n) (x) = G()f (n ) (x) + + G (n ) ()f(x) + G (n) (x y)f(y) dy. u () = u () = = u (n ) () = G() = G () = = G n 2 () =. p(d)u = G (n ) ()f(x) + p(d)g(x y)f(y) dy. G p(d)g(x) =, G (n ) () = p(d)u = f. f p(d)u = f G (n ) () =, p(d)g(x) = Green Titchmarsh injectivity theorem G.5. 2 [2] y + py + qy = f(x), y() = y () = Green G = G(x) y y(x) = G(x y)f(y) dy Duhamel 35 α, β (73) G(x) = eαx e βx α β 35 J.M.C.Duhamel ( , ) 834 Duhamel 8

81 α, β = a ± ib (a, b R, b ) α ax sin bx G(x) = e b (74) G(x) = xe αx = xe px/2 e αx e βx lim β α α β ( ) G.22 λ 2 + pλ + q = 2 α, β u(x) := G(x y)f(y) dy, G(x) := eαx e βx α β u u + pu + qu = f(x), u() = u () = u() = G() = u (x) = G(x x)f(x) + u () =. G () = u (x) = G (x x)f(x) + G (x y)f(y) dy = G (x y)f(y) dy = f(x) + G + pg + qg = u + pu + q = f(x) + G (x y)f(y) dy G (x y)f(y) dy. [G (x y) + pg (x y) + qg(x y)] f(y) dy = f(x) G.23 λ 2 + pλ + q = α u(x) := G(x y)f(y) dy, G(x) := xe αx u u + pu + qu = f(x), u() = u () = (xe αx ) G (74), (73) u(x) = G(x y)f(y) dy 8

82 G u() = u (x) = G()f(x) + G (x y)f(y) dy u () = G()f(). f u () = (75) G() = u (x) = G ()f(x) + G (x y)f(y) dy = u (x) + pu (x) + qu(x) = G ()f(x) + f [G (x y) + pg (x y) + qg(x y)] f(y) dy. (76) G () =, G + pg + qg = (75), (76) G C, C 2 { C e αx + C 2 e βx ( 2 α, β ) G(x) = (C + C 2 x)e αx ( α ). u() = u () = C, C 2 (73), (74) G.6 TO DO LIST [6] H 2 (?) (77) y + py + qy = f(x). (78) y + py + qy =. 82

83 H. H. ( 2 () ) λ 2 +pλ+ q = 2 α, β (79) y = Ae αx + Be βx (A, B ) (78) (a) A, B (79) y (78) (b) (78) A, B y = Ae αx + Be βx H.2 ( 2 (2) ) λ 2 + pλ + q = α (8) y = Ae αx + Bxe βx (A, B ) (78) (a) (8) y (78) (b) (78) A, B y = Ae αx + Bxe βx 2 2 ( ) H.2 ( ) H.2. y + ω 2 y = (ω ) y y y + ω 2 yy =. ( 2 (y ) 2 + ) 2 ω2 y 2 = 83

84 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 dy C2 y = ±ω 2 dx. 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 y = (A, B ) y = Ax + B (A, B ). H.2.2 H.3 D = d/dx α C, m N (8) (D α)y = e αx D(e αx ), (82) (D α) m y = e αx D m (e αx ). H.4 y = e px/2 z y + py + qy = z + ) (q p2 z =

85 y = e px/2 z y = p 2 e px/2 + e px/2 z, y = p ( p ) 2 2 e px/2 z + e px/2 z + ( p ) ( ) p 2 z + z e px/2 2 = 4 z pz + z e px/2 y + py + qy = 2 H.3 ( ) p 2 4 z pz + z e px/2 + ) ] (q p2 z. 4 = e px/2 [z + 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) =. y = e px/2 z z D 2 (e px/2 y) + (e px/2 y) =. y + py + qy = z + (q p2 4 ) z = (i) q p 2 /4 > ω = q p2 4 z + ω 2 z = z = A cos ωx + B sin ωx (A, B ). (83) y = e px/2 (A cos ωx + B sin ωx) (A, B ). (ii) q p 2 /4 = z = Ax + B z = (A, B ). (84) y = e px/2 (Ax + B) (A, B ). 85

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