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1 ( c Akir KANEKO) ).. m. l s = lθ m d s dt = mg sin θ d θ dt = g l sinθ θ l θ mg. d s dt xy t ( d x dt, d y dt ) t ( mg sin θ cos θ, sin θ sin θ). (.) m t ( d x dt, d y dt ) = t ( mg sin θ cos θ, mg sin θ sin θ), i.e. d x dt = g sin θ cos θ, d y dt = g sin θ sin θ x + y = l, x = l sin θ, y = l( cos θ) θ dx = l cos θ dθ dt dt, d ( ) x dt = l sin θ dθ + l cos θ d θ dt dt, dy dt = l sinθdθ dt d ( ) y dt = l cos θ dθ + l sin θ d θ dt dt )

2 ( ) cos θ, sin θ dθ dt l d θ dt = d x dt cos θ + d y dt sin θ = g sinθ cos θ cos θ g sin θ sin θ sin θ = g sin θ E = mv + mgy, E = (l m dθ ) + mgl( cosθ) ( ) dt E ml dθ dt d θ dt + mgl sinθdθ dt = ml dθ, dt. M, m Kepler 3 (.) µ = γm ( sec) (.496 m) 3 = M = m 3 kg sec M (.496 ) 3 ( ) kg = kg (Ris/Asir) kg ( 3 (^^;). ( ) ( ).3 λ v(x) = c e λx + c e λx v() = c + c =, v() = c e λ + c e λ = c = c. c (e λ e λ ) = c = c = λ e e λ =, e λ =, Euler λ = nπi, λ = n π (n =,,...) Euler v(x) = c e nπix c e nπix = c isin nπx

3 c = i v(x) = sin nπx. λ =, c + c x v() = c =, v() = c + c = c = c = λ =..4 (.3) dx = H u(x) g dt + u (x) u = y, u = dy dx (.3) θ c( cos θ) dθ dt = g dθ g dt = c, c( cos θ) + ( ) = cg ( cos θ) cos θ = cg( cos θ) sin θ cos θ cos θ θ = g c t + α t = x =, θ = α = (.3) x,y t ( ) g g g x = c c t sin c t, y = H c + ccos c t. () y = Ce x /.- { ( )} () y = C exp dx.- log x (3) y log y y = x (4) y = log(c x + C.-3 ).. () (5) y = Arccos Cex + Cex.. (). () y = ±x C log x. () Arctn y x log x + y = C.. () (3) y = xlog log C x.. (9) (x + y )3.3 () = C x y.3 () x + (y + ) = Ce Arctn((y+)/x).3. ().4 () y = x + C ex x.4 () y = x 3 3x + 6x 6 + Ce x.5. () (3) y = + Ce x /.4. () (4) y = + Ce x3 /3.4. (3) (5) y = + Ce /x.4. (7) (6) y = x e x + Ce x.4. (4) 3

4 .5 () y = x + Ce x.6 () y = ± + C x.7 () (3) x = e /3( ) y3 e y3 /3 dy + C.7 ().6 () x + xy = C.8- x 4 () 4 + y4 4 + x3 y 3 = C.8-.7 () e x (xy + y 3 )e x = C.9 () () e y x 3 e 3y + xe y = C.9 () 3.8 () y = cx + e c, y = x( log( x)). () y = cx + c, y = x 4.. () (3) y = cx c, y = ± x.. (4).9 () y = (± x + C), y =, y = x. () x = 3 p + C p, y = 3 p + C ( ), y =.. () p e p (3) x = p + e pdp + C, y = (ep ) { e p p + e pdp + C} ( ), y =.. (3). P(x)y = z { Q(x) + P (x) } P(x) y dz dx = P(x)dy dx + P (x)y = P(x) y + P(x)Q(x)y + P(x)R(x) + P (x)y { = z + Q(x) + P (x) } z + P(x)R(x) P(x) Q(x), P(x)R(x) R(x) P(x) z dy dx = y + Q(x)y + R(x) y + Q z dz dx = dy dx + Q = y + Q(x)y + R(x) + Q = z + R(x) + Q Q(x) R(x). () (.6) x = Aξ, y = Bη dy dx + y = B dη A dξ + B η = bx α = ba α ξ α B A = B = ba α 4

5 A, B, dη dξ + η = ξ α B =. A A = baα. A α+ = b. A = (b)/(α+), B = (b) /(α+) () (.7) = b/(α+) /(α+) = b/(α+) (α+)/(α+). dη dξ = d ( ) η (α n + 3)x αn+η = dx (α n + 3)x αn+η( xy + x dy dx ) = (α n + 3)x αn+η( xy + x ( y + bx αn ) ) = α b n + 3 η + η (x (α n + 3)x αn+ x y x = b α n + 3 η + ) (α n + 3)x αn+4 = α b n + 3 η + (α n + 3)ξ +/(αn+3) α n = n 4n + α n + 3 = + n 6n 3 4n = + n n 3 = 4n 4 n 3 = α n (A.) (.8). (.9) dy dx = (α n + )ξ +/(αn+) y dξ d ( ) y = (α n + )ξ +/(αn+) y ( ξη + ξ dη dξ + α ) n + b = y + bx αn dη dξ = α n + ξ 3 /(αn+) + b α n + ξ αn/(αn+) 3 /(αn+) y η ξ α n + bξ = α n + ξ 3 /(αn+) + = b α n + η α n + ξ 3 /(αn+) b α n + ξ 4( ξ 4 η + α n + b ξ 3 η + (α n + ) b ξ ) η ξ α n + bξ (A.) α n + = α, n + 3 α n + = α n+ + 3 (A.) ξ α n+ (.) (.9) (.7) (.7) (.7) x ξ, y η x = ξ αn+3, y = ξ η ξ η + b α n + 3 η = A = α n + 3 ξα n b α n + 3, B = α n dy dx + y = bx αn

6 = (α n + 3)B, b = (α n + 3)A dy dx + Ay = Bx α n dη dξ + (α n + 3)Bη = (α n + 3)Aξ αn n n +. (A.) dy dx + Ay = Bx αn ξ = x (αn+), y = ξ η + α n + B dη dξ α B n + η = α A n + ξα n+ (.9) = (.).. () y + y = x z = xy y z dz dx = y + xdy dx = y + x( y + x ) = x {xy (xy) + } = x (z z + ) dz z z + = dx x log x + c = dz (z )(z + ) = ( 3 z + z ) dz = 3 log(z + ) 3 log(z ) = 3 log z + z. z + z = Cx3, z = + Cx3 Cx 3, y = z x = + Cx3 x( Cx 3 ) ().4.3 y = cu + u x cv + v y = (cu + u )(cv + v ) (cu + u )(cv + v ) (cv + v ) = cu + u y(cv + v ) cv + v c. c(yv u ) = u yv, c(y v u + yv ) = u yv y v (y v u + yv )(u yv ) = (yv u )(u yv y v ), y {v (u yv ) + v (yv u )} = (yv u )(u yv ) ( u + yv )(u yv ) (v u u v )y = (v v v v )y + (u v + v u v u u v )y u u + u u y = v v v v v u u v y + u v + v u v u u v v u u v y + u u u u v u u v 6

7 u, u, v, v x P, Q, R c x c = v y + u v y u = ( v y v y + u )(v y u ) ( v y + u )(v y + v y u ) (v y u ) (u v v u )y + (v v v v )y + (v u + u v v u u v )y + u u u u = y.4 () y = 3 x 9 + c e x + c e 3x 3.. () () y = + c e x + c e x 3.. () (3) y = 3 cos x + c cos x + c sin x 3.. (3) (4) y = cos x + c e x + c xe x 3.. (4) (5) y = x + c e x + c e x 3.. (5) (6) y = e x + c e x cos x + c e x sin x 3.. (6).5 dy dx = dy / dx = dθ sin θ dθ cos θ θ, ϕ sinϕ = sin θ ( cos θ) + sin θ = v = ( dx dt, dy dt ) = (dx dθ, dy dθ )dθ dt sin θ cos θ = sin θ sin θ = ( cos θ,sinθ)dθ dt ( ) d v = (sin θ,cos θ) dθ + ( cos θ,sin θ) d θ dt dt dt = cos θ φ ((θ sin θ), ( cos θ)) mg, (cos ϕ,sin ϕ) = (sin θ,cos θ ) α = (sin θ sin θ + cos θ cos θ ) ( dθ dt ) + {( cos θ)sin θ + sin θ cos θ }d θ dt = cos θ ( dθ dt ) + sin θ d θ dt 7

8 Newton mα = mg sin ϕ = mg cos θ dθ dt, s, ds = dx + dy = ( cos θ) + sin θdθ = sin θ dθ ds dt = sin θ dθ dt, d s dt = g cos θ s = s ds = θ π d s dt = sin θ d θ dt + cos θ ( ) dθ = α dt sin θ [ dθ = 4cos θ ] θ = 4cos θ π d s dt = g 4 s..6 λ + λ + b =.. y = B µ + µ + b eµx + c e λ x + c e λ x y() = y () = = c = λ λ λ + B µ + µ + b + c + c, = B(µ λ ) (λ λ )(µ + µ + b), c = λ λ λ Bµ µ + µ + b + c λ + c λ B(µ λ ) (λ λ )(µ + µ + b), y = λ e λ x λ e λ x λ λ + B λ λ (λ λ )e µx + (µ λ )e λx (µ λ )e λx µ + µ + b µ λ (λ λ )e µx + (µ λ )e λ x (µ λ )e λ x µ + µ + b = (λ λ )(e µx e λ x ) + (µ λ )e λ x (µ λ )e λ x (µ λ )(µ λ ) xe λ x + eλ x e λ x λ λ y = λ e λ x λ e λ x λ λ + Bxeλ x λ λ B(eλx e λx ) (λ λ ) 8

9 λ + λ =, λ λ = λ + y = B λ + xeλ x + c e λ x + c e λ x c = λ λ λ + = c + c, = B λ + + c λ + c λ B (λ + )(λ λ ), c = λ λ λ B (λ + )(λ λ ), B y = λ + xeλ x + λ e λx λ e λx + B(eλx e λx ) λ λ (λ + )(λ λ ),.7, y + y + ω y = Be iωx y = Be iωx ω + iω + ω + c e ( +iω)x + c e ( iω)x = Beiωx iω + c e ( +iω)x + c e ( iω)x y() =, y () = = B iω + c + c, = Biω iω + c ( + iω) + c ( iω) c = iω iω B( + iω) (iω), c = + iω iω + B (iω) y = Beiωx iω + ( iω)e( +iω)x + ( + iω)e ( iω)x + iω B (iω) { ( + iω)e( +iω)x + e ( iω)x } = + ( iω)e( +iω)x + ( + iω)e ( iω)x iω + B (iω) {iω ( + iω)e( +iω)x + e ( iω)x } = iω B e ( +iω)x B (iω) {e( +iω)x e ( iω)x }, e ( +iω)x = ( iω)x x = x Tylor e( +iω)x = + e x e x cos ωx ω = 9

10 x x.8 () 3 ex + x x + e 8 x + c e x + c e x 3.. (7) () e 3x ( x x + ) + c e x + c e x 3.. (8) (3) log(cos x)cos x + xsinx + c cos x + c sin x 3.. (9) (4) x3 (log x) + c x 3 + c x 3 log x 3.. (4) (5) 4 + c cos(log x) + c sin(log x) 3.. (5).9 d 4 y dx 4 = dx d { = p ( d 3 ) y dx 3 = dy dx dy d { p d p p d 3 p dy 3 + pdp dy = p 3 d 3 p dy 3 + dp d ( p dp 4p dy dy + p dy. d y dx = f(y) dy dx ( ) dp } dy + p dy d ( ) p dp 3 ( dp dy + + p dy dy ) 3 d y dx dy dx = f(y)dy dx f(y) F(y) ( ) dy = F(y) dx ) d p dy } x. ( ) dy = f(y)dy + C dx.. (.4) P(x)y = u u ( u = y Py ) ( u ) ( u ) Qy R = P + Q R Pu Pu Pu = Puu P(u ) P uu P u P(u ) P u + PQuu P u P Ru P u = Pu {u P P u Qu + PR}

11 y = u /Pu u = Puy, u = Puy (P u + Pu )y = Puy + P uy P uy, u Py + P y P y PSy + T =, y Py + ( P P + S)y T P = 3 3. f(x) = x x = Lipschitz, x K x K x x = - Hölder x y x y x + y x y x y x y / x + y x + y x y x + y x + y x y x y /, x f(x) = log x,, x = x = Hölder. x x, x log x, x = log x K x α α > x α log x x 3. [] von Koch t =.i i...i n... f(t) n 3 T i i...i n n+ t t < n f(t ) f(t ) n n 3 t t n,n,, n + n + t, t, t t n ( ) n 3 3 n f(t ) f(t ) ( 3 ) n = 3( 3) (n+). n+ t t n + log t t log

12 n+ t t < n log f(t ) f(t ) log 3 (n + )log 3 log 3 + log t t log f(t ) f(t ) 3 t t log 3/log log 3 n f [,] -Hölder log log 3 T T T T T T f(t) R f(t) x,y 3 x R Euclid Hölder 3 ( 3.) 3 T i i i n t =.i i i n t =.i i i n n x Hölder y 3 t f(t) t n+, f(t) y 3 3 n Hölder 3.3 f(x) = { xsin x, x,, x = C Lipschitz f(x) f() x = x Lipschitz Lipschitz x = nπ ε, x = nπ + ε, f(x ) = (nπ ε) sinε, f(x ) = sin ε, (nπ + ε) x x = ε 4n π ε, f(x ) f(x ) = 4nπ sin ε 4n π ε, f(x ) f(x ) K x x 4nπ sin ε Kε, n π ε sin ε n

13 Hölder Hölder { x α sin, x f(x) = x,, x =. Hölder x = α-hölder x x x x = ε 4n π ε (, ) f(x ) f(x ) = sin ε nπ ε α + nπ + ε α = sin ε nπ + ε α + nπ + ε α 4n π ε α f(x ) f(x ) K x x α sinε( nπ + ε α + nπ ε α ) K(ε) α n ε nπ + ε α + nπ ε α = (nπ) α( + ε nπ α + nπ ε α) = (nπ) α ε n. 3.4 ([5]), x X x n = T n x, n =,,,... m > n dis(x m,x n ) = dis(t m x,t n x ) dis(t m x,t m x ) + dis(t m x,t n x ) T dis(t m x,t m x ) + dis(t m x,t m x ) + + dis(t n+ x,t n x ) λ m dis(tx,x ) + λ m dis(tx,x ) + + λ n dis(tx,x ) = (λ m + λ m + + λ n )dis(tx,x ) n= λ n < m,n x n = T n x X Cuchy x X dis(tx,ty) λ dis(x,y) T Tx n Tx x n+ = Tx n x = Tx x T. x, y dis(x,y) = dis(tx,ty) = = dis(t n x,t n y) λ n dis(x,y) n λ n dis(x,y) =, x = y n = dis(x m,x ) = dis(t m x,t x ) (λ m + λ m + + λ )dis(tx,x ) m dis(x,x ) λ n dis(tx,x ) n= 3

14 Tϕ := c + f(t,ϕ(t))dt f(x,y) x = ϕ Tϕ T X (Tϕ)(x) (Tψ)(x) {f(t,ϕ(t)) f(t,ψ(t))}dt K K ϕ ψ t pdt = ϕ(t) ψ(t) t p dt K p x p ϕ ψ K p p ϕ ψ δ := K p p < X = C[,] p.78 [,] ϕ (x) x δ, ϕ(x) = ϕ (δ) + δ f(t,ϕ(t))dt δ > δ x δ + δ ϕ (x) { ϕ (x), x δ, ϕ(x) = ϕ (x), δ x δ + δ, δ x δ + δ ϕ(x) = ϕ (δ) + δ δ f(t,ϕ(t))dt = c + f(t,ϕ(t))dt + δ f(t,ϕ(t))dt = c + f(t,ϕ(t))dt x = f(x,y) x C ϕ (x) = f(x,ϕ(x)) δ > x δ f(x,y) y Lipschitz. ϕ(x) := mx x {e L x ϕ(x) } λ < δ > K p δ p = λ x e Lx (Tϕ)(x) (Tψ)(x) e Lx {f(t,ϕ(t)) f(t,ψ(t))}dt K e Lx ϕ(t) ψ(t) t p dt K mx t x e Lt ϕ(t) ψ(t) e L(x t) t p dt ϕ ψ K e L(x t) t p dt 4

15 x δ λ ϕ ψ x > δ mx t δ e Lt ϕ(t) ψ(t) K δ e L(x t) t p dt + mx δ t x e Lt ϕ(t) ψ(t) K λmx δ t x e Lt ϕ(t) ψ(t), mx δ t x e Lt ϕ(t) ψ(t) K δ p δ δ e L(x t) t p dt e L(x t) dt = K δ p L ( e L(x δ) ) mx δ t x e Lt ϕ(t) ψ(t) L K δ p λ L λmx δ t x e Lt ϕ(t) ψ(t) e Lx (Tϕ)(x) (Tψ)(x) λ mx t e Lt ϕ(t) ψ(t) x x mx x e Lx (Tϕ)(x) (Tψ)(x) λ mx t e Lt ϕ(t) ψ(t) x T 3.8 x mx{ f (x),..., f n (x) } f k (x) Hölder < p < p + = q f(x) = f (x) + + f n (x) ( f (x) p + + f n (x) p ) /p ( q + + q ) /q = n /q f(x) p n /q ( f k (x) p + + f k (x) p ) /p = n /q n /p f k (x) = n /q+/p mx{ f (x),..., f n (x) } = n f f(x) = mx{ f (x),..., f n (x) } = f k (x) f (x) + + f n (x) = f(x) x 3.9 () x, y f( y) f( x) = d f( x dt + t( y x))dt = d f dt ( x + t( y x))dt ( y x) f C M, Schwrz M y x f Lipschitz M Lipschitz () f x y y = P n (.5 n,), P n+( n.5,) + P n P n+ =.5 n(n + ) f(p n ) f(p n+ ) = 4 = K f(p n ) n+ ) K P n P n+ 5

16 3. y = f(y) := { y α, y,, y < f(y) α-hölder y > y > f(y ) f(y ) = αy α yα (y y ) α < α <,b ( + b) α α + b α b α t = b (x + ) α x α + y α = (y y + y ) α (y y ) α + y α y > y f(y ) f(y ) = f(y ) = y α = (y y ) α y dy = f(y) y α = dx, α y α = x C. y = {( α)(x C)} /( α) /( α) > x = C x y = α = 3. { } ϕ(x) C + K dt C + K ϕ(t )dt x = C + CK( x) + K dt t x C + CK( x) + C K ( x) + dt t dt! t dt t ϕ(t )dt + + C Kn ( x) n n! t n ϕ(t n )dt n Gronwll ϕ(x) M n 3., g(x;λ) x, λ Λ R N G(λ) = mx g(x;λ) λ. µ Λ y x ε > g δ y > x, λ Λ x y < δ y, λ µ < δ y g(x;λ) g(y,µ) < ε, y δ y - V y y Heine-Borel V y,...,v yk 6

17 δ = min{δ y,...,δ yk } x,y x y < δ λ µ < δ g(x;λ) g(y,µ) < ε y V y,...,v yk, V y x y x y + y y < δ + δ y δ y g(x;λ) g(y,µ) g(x;λ) g(y,µ) + g(y;µ) g(y,µ) < ε + ε = ε g(x;µ) x x = x µ x x µ < δ, λ µ < δ, g(x;λ) g(x µ,µ) < ε g(x;λ) > g(x µ,µ) ε G(λ) = mx g(x;λ) > g(x µ,µ) ε = mx g(x;µ) ε = G(µ) ε x x x λ µ < δ g(x;λ) g(x,µ) < ε g(x;λ) < g(x,µ) + ε mx g(x;µ) + ε x G(λ) = mx x G(λ) G(µ) = mx x g(x;λ) < mx g(x;µ) + ε = G(µ) + ε x g(x;λ) mx g(x;µ) < ε x G(λ) = mx g(x;λ) λ 4 x 4. y + y + by = f(x) (,b x.) y = y, y = y { y = y, y = y by + f(x) ( ) ( )( ) ( ) y y y = b y + f(x), t ϕ,ϕ, t ψ,ψ ( ) ( )( ) y y = ϕ ψ c ϕ ψ c t (c,c ) ) ( y y ) = ( ϕ ψ ϕ ψ c, c ) (c c ) + ( ϕ ψ ϕ ψ ) ( c c = ( ) ϕ ψ ( ) ( ) c ϕ ψ c = f(x) ( )( ) ( ) ( ) ϕ ψ c ϕ ψ ( ) b ϕ ψ c + c ϕ ψ c { c ϕ + c ψ =, c ϕ + c ψ = f(x) 7

18 ϕ = ϕ, ψ = ψ, ϕ, ψ c ϕ + c ψ = f(x) 4. () () λ (3) n Euclid ( + b ) + + ( n + b n ) + + n + b + + b n n n 4.3 () e ta = 3 et + 3 e4t 3 et + 3 e4t 3 et + 3 e4t 3 et () e4t ( 7e t 6e t 3e t 3e t 4e t + 4e t ) () e ta = 6e t + 6e t e t + 3e t 4e t 4e t 6..() 6e t 6e t 3e t 3e t 3e t + 4e ( t (t )e t (t )e t e t (t 3)e t 3e t ) (3) e ta = (4) e ta = 3te t 3(t )e t + 4e t (3t 6)e t + 6e t ( te t (t )e t e t (t )e t 3e t e t te t te t ) (6t + )e t 6te t 6..(5) 6te t (6t )e t e t te t 4te t (5) e ta (4t + )e = t 4te t 4te t (4t )e t. 6..(4) te t te t e t (6) e ta = (7 t + 3t )e t ( t + t)e t ( 3 t + t)e t ( 35 t 6t)e t ( 5 t + t )e t ( 5 t t)e t (4t + 5t)e t (t + 3t)e t (6t + 4t + )e t 6..(3) 6.() µ m x m e µx µ Jordn m Jordn (c ) 4.5 S AS = Λ Jorndn Se xλ c c 3 ( ) () A =. λ 3 + λ 4λ 4,,,. ( ) S = S 6 = 3 ( e x ) e xa = S 8 e x e ( x e x )( ) c S 8 e x c e x y 3 = c e x c e x. ( 6.. (9) ) () A = c S AS = ( ). S S y = c e x + c e x + c 3 e x, y = c e x + c e x c 3 e x,. λ 3 λ + λ,, ±i. S = 8

19 ( ) i + i i i ( e x S e ix, S = +i +i 4 4 i 4 +i 4 i 4 +i 4 ; S AS = ( ) i. i )( ) c c e xa e ix c 3 ( e x ) cos x sin x sin x S e ix S = e ix ( ex + cos x + sinx) (ex + cos x sinx) (ex cos x + sin x ( ex + cos x sinx) (ex cos x sinx) (ex + cos x + sinx) ) ( c c c 3 y = c cos x c sinx + c 3 sin x, y = c ( e x + cos x + sin x) + c (e x + cos x sin x) + c 3 (e x cos x + sin x), y 3 = c ( e x + cos x sin x) + c (e x cos x sin x) + c 3 (e x + cos x + sin x). ( ) 6.. (5) (3) A =, λ 3 3λ, ( ),. 5 Jordn ( ) 3 S = 3, S = 9 5 ( ) S AS =. 9 6 ( e x xe x )( ) c S S e x c y = c e x +c (x+)e x +c 3 e x, e x c 3 y = c e x c xe x + c 3 e x, y 3 = 3c e x + c (3x + )e x + 6c 3 e x. 6.. (6) 4.6 () S e xa y = c e x +c e x +c 3 e x, y = c e x +c e x c 3 e x, y 3 = c e x c e x = c + c + c 3..., = c + c c 3..., = c c c 3 =, c + c =. 3 c =, c =. y = e x + e x, y = e x e x, y 3 = e x. () e xa c =, ( c c c 3 c =, c 3 = y = cos x sin x, y = ( e x + cos x + sinx) (ex cos x + sin x) = 3ex + 3 cos x + sin x, y 3 = ( e x + cos x sinx) (ex + cos x + sin x) = 3ex + cos x 3 sin x. (3) = c + c + c 3..., = c + c 3..., = 3c + c + 6c = c + 5c 3. c 3 = 3, c =. 3 c = 3. y = (3x )e x 3 ex, y = (3x 3 )e x 3 ex, y 3 = (9x + )e x e x. 4.7 (, ) > + ) 9

20 (, ) > + = 3 +, > 3, > (+) 3 + > ( 3 + ) (+) (+) ( 3 + ) = ( + )( ) ( 3 + ) = + ( + ) = ( ) ( ) ( ) + z () + z = z ( + ) + + > + > + + > > > (+)( ) = > > > ( ) ( ) ( ) y = e x + e x, y = e x + e x, y 3 = e x. z (3) z = 5 z 3 > + > ( + ) ( + ) > 4 ( + ) = ( +)( )+( 5) > 5 4 ( + ) 3 3 > (5 4) ( 3 3 ) ( 3 3 ) ( +)(5 4) > (5 4) 3 ( 3 3 ) 5 4 > (5 4) ( 4+) ( 3 3 )

21 3 3 = ( + ) ( ) z = 4 ( + ) ( ) = ( + ) ( ) = 3 ( + ) + 7 3( + ) 3( ) 3xe x + 3 7e x 3 ex, z = 5 4 ( + ) ( ) = 3 ( + ) + 3( + ) 3( ) 3xe x + 3 e x 3 ex, z 3 = 4 + ( + ) ( ) = 9 ( + ) + + 9xe x + e x e x. 4.6 (3) 4.9 () y = c cos x + c 3 sin x, y = { c e x + c (sin x + cos x) + c 3 (sin x cos x)}, y 3 = { c e x + ( ) c ( sin x + cos x) + c 3 (sin x + cos x)}. Jordn J = i, i ( ). 6.. (5) (3) y = c x e x c e x +c 3 e x. y = c ( x + e 3 x + 3 c e x c 3 e x. y 3 = c (x+ 3 )e x +c e x 3c 3 e x. ) ( Jordn. 6.. (6) 4. () y = 3 x 9 + c e x + c e 3x, y() = 9 + c + c =, 5 9 y() = c e + c e 3 =. (e e 3 )c = 5 9 9e3 c 9e = 3 e, c = e 3 y = x e {( e )e 3x + ( 9e )ex }. e 3 e e e 3, () y = +c e x +c e x, y = c e x c e x., y () = c c =, y () = c e c e =. c = c. c (e e ) =. c =, c =. y = (3) y = 3 cos x + c cos x + c sin x., y() = 3 + c =. c = 3. y(π) = 3 c. c =. 3 (4) y = cos x + c e x + c xe x. y = sinx + c e x + c (x + )e x., y() = + c = y(π) = + c e π + c πe π. (e π )c + πe π c =. y () = c + c = y (π) = c e π + c (π + )e π. (e π )c + {(π + )e π }c =. (ff π )c =, c = e π. c = (π + )eπ (e π ). y = cos x + (π + )eπ (e π ) e x e π xex. (5) y = x + c e x + c e x. y = + c e x c e x., y( ) = +c e +c e =, y () = +c e c e =. c (e+e )+c (e e ) =. c = e+e c e e. ( e e e+e e e )c =. c = e e e + e. c = e + e e + e. y = x + e e e + e ex e + e e + e e x. (6) y = e x + c e x cos x + c e x sin x., y( π) = e π c e π =, y(π) = e π c e π =. c = y = e x ( + cos x + csin x). 4. () λ = n π 4, n =,,,..., y = e x sinnπx 5.5. () () λ = 9 4 n π, n =,,,... y = e x/ (nπ cos nπx sin nπx), y = 5.5. ()

22 (3) n π, n =,,... sin nπx. y + y = λy y = ( λ)y 4.4 () (4) λ = 4n 4ni, n =,,,... ( ), y = e nix. ( ) 5.5. (9) (5) λ = (n + ) π, n =,,,..., sin n (6) λ = n, n =,,,..., 4 y = ex sin nπ π(x + ) 5.5. (6) (x + π) 5.5. (7) 4. π + nπ < t < π + nπ, n =,,... cosh t cos t = t n,t n, n =,,... λ n = t4 n 4, n =,,.... y = sint n e tnx/ + (cos t n e tn )e tnx/ + (cos t n sin t n e tn )sin t nx (cos t n + sin t n e tn )cos t nx [ 5.8-] 4.3 r(x) ( d r(x) dx p(x) dy dx ) + q(x) r(x) y = λy r(x) R(x) r(x) > R(x) X = R(x) C, x = R (X) C [R(),R(b)] p(r (X)) r(r (X)) p, q(r (X)) r(r (X)) Y (X) = y(r (X)), dx d ( P(X) dx dy ) + Q(X)Y = λy q P(X) > p(x) C P(X) C r(x) C Sturm-Liouville p(x) r(x) 4.4 λ ϕ(x) (Lϕ,ϕ) = ( dx d p(x) dx d ϕ,ϕ) = (p(x) d dx ϕ, d ϕ) dx p(x) [,b] p(x) M > M (Lϕ,ϕ) M ϕ. (λrϕ,ϕ) = λ(rϕ,ϕ) r(x) [,b], (rϕ,ϕ) = λ M ϕ (rϕ,ϕ) > b r(x)ϕ(x) dx

23 4.5 L λ n, ϕ n (x) λ = λ k L k Lu = λu+f f = n= f nϕ n (x) u = n= u nϕ n (x) λ n u n ϕ n (x) = λ k u n ϕ n (x) + f n ϕ n (x) n= n k, n = k n= u n = f n λ n λ k = f k f k = (f,ϕ k ) u k, 5 5. () y = x + c x n n! [ 7.. ()] n= x n+ () (n + )!! + c e x / [ 7.] n= (3) c x 3n 3 n n! [ 7.. ()] n= n= (4) + c x n n n! [ 7.. (3)] (5) c n= x n n! n + n n! n x n [ 7.. (5)] n 5. () nc n = c n k c k (n ). c y = c n+ x n c = c x k= [ 7.. ()] () c c = c, c () c = c c = c 3 c, c 3 = 3 (c c +c ) = 3 {c (c 3 c )+(c ) } = c c + 3, c 4 = 4 (c c 3 +c c ) = 4 {c (c c + 3 )+(c )(c3 c )} = c c3 + 3 c, c 5 = 5 (c c 4 + c c 3 + c ) = 5 {c (c c3 + 3 c )+(c )(c4 4 3 c + 3 )+(c3 c ) } = c 6 c c 5. [ 7.. ()] (3) c c = c c = c 3 + n=, nc n n = c n k c k (n 3) c 3 = 3 ( c (c 3 + ) + c4 ) = c4 3 c, c 4 = 4 { c ( c 4 3 c ) + ( c )(c3 + )} = c5 + 5 c, c 5 = 5 { c (c c ) + (c4 + 3 c )c + (c3 + ) }. [ 7.] (4) c c = c, c = (c c + c c ) = c c = c 3. c 3 = 3 { + (c c + c + c c )} = 3 + c4. (n + )c n+ = c c n + c c n + + c n c. c 4 = 4 (c c 3 + c c ) = ( 3 c +c 5 +c5 ) = 6 c +c 5, c 5 = 5 (c c 4 +c c 3 +c ) = 5 ( 3 c +c6 + 3 c +c6 +c6 ) = 5 ( c +5c6 ) = 5 c +c6. [ 7.. (5)] 3 k= n=

24 ( ) n ( ) n 5.3 () y = c (n)! xn + c (n + )! xn+ + ω n= n= () y = c (n)! xn + c (n + )! xn+ x [ 7.3. ()] n= (3) y = c n= n n= (n )!! xn + c n= y Ry = RC log n! xn+ [ 7.3. (3)] 5.4 y = C ( x R )( y R ) ( R) y dy = C x dx, y ( y R R = C log R x ), ( R x ), (y R) = R RC log y R = ± R RC log ( R x ) n= ( ) n (n + )! ωn+ x n+ [ 7.3. ()] ( R x ) R ( log + R r ) = C R, r = R(eR/C ) R C R x y R R C Tylor y C( + y) = x R dy + y = C x dx, R ( log( + y) = C log R x ) (, y = + R x ) C x < R R C 5.5 (4.) ρ > R ρ ϕ,..., ϕ n ϕ (n) + ϕ (n ) + + n ϕ =,..., ϕ (n) n + ϕ (n ) n + + n ϕ n = j Crmér n j+ = W j W(ϕ,...,ϕ n ),, ϕ,..., ϕ n Wronski, ϕ (n) j. ρ ϕ n (n) 4

25 f(x) f(x) ρ R R 5. ( ) R () f(x), g(x) ρ, f(x)g(x) ρ () f(x) ρ f(x) x x < ρ f(x) ρ y = c. y = x y c ( ) x. y = x c y (x ) x = c x + c, y = c + c (x ). + x y + y x = x y = + x c c = () n := ( + ) ( + n ) y = () n (b) n n= x n (c) n n! c y = c ( c + ) n (b c + ) n n= x n+ c c ( c) n n! y = ( ) n n!x n y = ( ) n nn!x n, x y = n= ( ) n nn!x n+ = n= n= n= ( ) n (n+)!x n+ ( ) n n!x n+ = y + x (xy x) = (x+)y + n= x y + (x + )y =. dy y = x + x, log y = log x + x + C, y = c x e/x. c xe /x =, c = e /x x dx + C ( y = e/x x e /x ) x x dx + C. x = C = e /x x dx = xd(e /x ) = y = e/x x [xe /x] x x 5 e /x x dx. e /x dx = xe /x x d(e /x )

26 = {x x + x ( ) n n!x n+ e /x } ( ) n (n + )!x n e /x dx. y = x + x + + ( ) n n!x n ( ) n e/x x (n + )!x n e /x )dx (n + )!x n+ e /x (n + )!x n = O(x n ). x +. ( O(x n+ ) n.) 5.9 Φ(x)Φ(x) = I Φ(x) Φ(x) Φ(x) + Φ(x){Φ(x) } = O {Φ(x) } = Φ(x) Φ(x) Φ(x) h n (x) = ( ) n dx dn n e x, H n (x) = h n (x)e x H n n, x n. (.) h n (x) = h n+(x) = H n+ (x)e x (A5.), h n (x) = H n (x)e x H n (x)e x xh n (x)e x (A5.). H n+ = xh n H n (A5.3). Leibniz n+ dn+ n+ dn h n+ (x) = ( ) = ( ) dx n+e x dx d n n+ dn dx e x = ( ) dx n ( xe x ) = x( ) n dn dx n e x + n( ) n dn dx n e x = xh n (x) nh n (x). H n+ = xh n nh n (A5.4) H n = nh n. H n = 4n(n )H n (A5.5) (A5.6) 6

27 . (A5.4) xh n = H n+ + nh n. ( x / ) d dx e x / = d dx ( xe x / ) = (x )e x / (A5.7) (A5.8) Leibniz, (A5.5), (A5.6), (A5.7) ( d dx + x )(H n (x)e x / ) = { H n (x) + xh n (x) (x )H n (x) + x H n (x)}e x / = { 4n(n )H n (x) + 4nxH n (x) + H n (x)}e x / = { 4n(n )H n (x) + n(h n + (n )H n ) + H n (x)}e x / = (n + )H n (x)}e x / ψ n (x) = H n (x)e x / n + ψ n (x) dx = = H n (x) e x dx = H n (x)( ) n dn dx n e x dx H n (x)(h n (x)e x )dx H n (x) n x n = ( dn dx n H n (x))e x dx = n n!e x dx = n n! π. ψ n (x). H =, H = e x d dx e x = x, H = e x d dx e x = 4x, H 3 = e x d 3 dx 3e x = 8x 3 x (A5.6) n H n (x)e x / ( d dx + x )e x / = ( x + + x )e x / = e x /, ( d dx + x )(xe x / ) = ( (x )x + x + x 3 )e x / = 3xe x /, ( d dx + x )((x )e x / ) = { (x )(x ) + x x + x (x )}e x / = 5(x )e x /, ( d dx + x )((x 3 3 x)e x / ) = { (x )(x 3 3 x) + (3x 3 ) x 6x + x (x 3 3 x)}e x / = 7(x 3 3 x)e x / 7

28 6 6. y x, y = g(y) x = c z = z(x) x y(x) z(x) g(y) y y x = z(x) y = g(y). f(x, y) g( y ) d y dx d y dx f(x, y) g( y ) g( y ) x x = d y x g( y ) dx dx y(x) c f(x, y(x)) dx x g(y) dy y + x x y g(y) = K y Osgood. Lipschitz 7 7. (6) (x,y ) y > x y + (x,y ) 3, x y <. x y y x x, x y x y < x x, dy dx = x y x y, y x dy dx x > ( log y x x y + log y x ) x y + x x x x = y y x x, x < ( ) y Arctn y Arctn x x x x x, y + x y x log y + x x y, x x >, x x + y, x =, x ( π y Arctn ), x <, x x 8

29 + 7., K ϕ(x) ϕ(x) ψ(x) ϕ(x) ϕ() + K ϕ(t)dt x ϕ(x) ϕ()e K(x ) x ϕ(x) ϕ() + K { ϕ() + K t = ϕ(){ + K(x )} + K dt ϕ(t )dt }dt t {K(x )} ϕ() { + K(x ) +! t tn + K n+ ϕ(t n )dt n ϕ(t )dt + + {K(x } )}n n!, [,b] ϕ(x) M ϕ(t n ) M {K(x )} n+ Gronwll M (n + )! n n ϕ(x) ϕ()e K(x ) [,b] b x 7.3 () x y y Lipschitz Lipschitz () y = ±x y < x y > x y = x x > x < (3) 7. y = x y(x y ) = y( x y + y x ). x y(x y ) =. x + 4y y > 4 y y < < x < + + 4y 4 y. y x = ± y y = ±y + y y = ±x y < x = y = y + y y y = ±x y x (4) () y > x..., y = x, x > x > y... 4 (b) x < y... y = x, x <, y = x, x < 3, 4 (c) y < x... 3 x = + + 4y 4, y y < ( ) y = x, x > 4 9

30 (d) 4 x = + + 4y 4, y > y x = + + 4y 4, y y (e) 3 4 x = + + 4y 4, y y 7. (8) (5) ( ) x x [6] y = sin y dy siny = dx, x + C = dy sin ydy dcos y siny = sin y = cos y = cos y log + cos y cos y = cex + cex, y = Arccos cex + ce x x y = (n + )π, n Z y = log( + x ), x y = log( + x ) dx+c log( + x ) dx+c+ log x dx = log( + x ) dx+c+ log x dx x + sin y x nπ, n Z y < c π x = log( + x x ) < y < π π < y < π y = π. x y = π y = π ε >, π + ε y π ε x y π + ε x π + ε y π ε siny sinε log( + x ) sin ε x x := exp sinε y sinε. y y(x ) sin ε(x x ) 3

31 y = π. y π y = π y = y = 3π y = π y = π y = 3π y = π y = π x = c c 7. (8) π π 7.5 y y = 3y y = Ce 3t x = c e t y = 3y + c e t. (e 3t y) = c e t, e 3t y = c e t + c y = c e 3t + c e t. O(e t ) c = y O(e 3t ) x 7.6 () 7. (), dθ = xy yx dt x + y = { ( r x y log x ) ( y x + y )} = r log r log r dr dt = xx + yy r = r { ( x x + r = Ce t, log r = t c, dθ = dt t + c, θ = θ + y ) ( + y y log r t t + c dt log x )} = r r t 7.7 dr dt = xx + yy r = { ( ) ( } r x x + O( r (log r) ) + y y + O( r (log r) ) dθ = xy yx dt r = { ( ) ( } r x y + O( r (log r) ) y x + O( r (log r) ) 3 = r + O( r (log r) ), = O( (log r) )

32 , ( + ε)r dr dt ( ε)r Ce (+ε)t r Ce ( ε)t, r dθ dr c r(log r), θ c r ( r r(log r) dr = c log r log r ) t, r 7.8 y > y < δ > ( ) δ δ y = δ (x,δ) δ x x + x t = (x + (t),y + (t)), (x (t),y (t)) δ x + t = T (δ,m), ε > x = δ, y = m ε, T y = δ x + x + x, δ > δ y > y x x t x x + = x x + x >. y. µ < µ, µ > x = λx+g(x,y), y = µy +h(x,y), λ,µ >, g,h C o( x + y ) g, h x,y ε > δ >, δ (x,y ), (x,y ) g(x,y ) g(x,y ) = g(x,y ) g(x,y ) + g(x,y ) g(x,y ) = g x (ξ,y ) x x + g y (x,η) y y ε x x + ε y y h(x,y) (x +,δ), (x,δ) t = (x (t),y (t)), (x (t),y (t)) y = µy + h(x,y ), y = µy + h(x,y ), (y y ) = µ(y y ) + h(x,y ) h(x,y ) µ(y y ) ε(x x ) ε y y (y y ) µ(y y ) ε(x x ) ε y y 3

33 p.69, 6.5 y y (y y ) y (t) y (t) (y (t) y (t)) = min{(y (t) y (t)),(y (t) y (t)) } x min{ µ(y (t) y (t)), µ(y (t) y (t))} ε(x (t) x (t)) ε y (t) y (t) = µ y (t) y (t) ε(x (t) x (t)) ε y (t) y (t) = (µ + ε) y (t) y (t) ε(x (t) x (t)) (x (t) x (t)) = λ(x (t) x (t)) + g(t,x (t),y (t)) g(t,x (t),y (t)) λ(x (t) x (t)) ε(x (t) x (t)) ε y (t) y (t) = (λ ε)(x (t) x (t)) ε y (t) y (t) λ ε µ + ε (δ ε, λ > µ, x > x {(x (t) x (t)) y (t) y (t) } (λ ε)(x (t) x (t)) (µ + ε) y (t) y (t) (µ + ε){(x (t) x (t) y (t) y (t) } {(x (t) x (t)) y (t) y (t) )e (µ+ε)t } t = x (t) x (t) = x + x, y (t) y (t) = t (x (t) x (t)) y (t) y (t) )e (µ+ε)t x + x (x (t) x (t)) y (t) y (t) (x + x )e (µ+ε)t t x (t) x (t) y (t) y (t) t ( x (t) x (t) > t t = x (t) x (t) > x > x y, y ) (x,y ) (x,y ) λ µ λ,µ 33

34 ( ) d y y dt x x = { µ(y (A7.) y ) + h(x,y ) h(x,y )}(x x ) {λ(x x ) + g(x,y ) g(x,y )}(y y ) (x x ) y t y t x x y y (x x ) (A7.) ( ) d y y y (λ + µ 3ε) y + ε dt x x x x { y y x x e (λ+µ 3ε)t} εe (λ+µ 3ε)t y y x x y y x x e (λ+µ 3ε)t ε λ + µ 3ε {e(λ+µ 3ε)t } y y x x ε λ + µ 3ε (x x ) y y y y y y { y y x x e (λ+µ 3ε)t} εe (λ+µ 3ε)t y y ε x x λ + µ 3ε y y y y y y ε x x λ + µ 3ε ε y y x x (x x ) = λ(x x ) + g(x,y ) g(x,y ) λ(x x ) ε(x x ) ε(y y ) (λ ε)(x x ), {(x x )e (λ ε)t } (x x )e (λ ε}t x + x x x (x + x )e (λ ε)t λ ε > ε t x x y y x x y > 7.9 x = ϕ(ξ,η), y = ψ(ξ,η), ( dx ) dt dy = dt ( ϕ ξ ψ ξ ) ϕ (dξ η ψ dt dη η dt ) = ( ) f(ϕ,ψ) g(ϕ, ψ) (dξ dt dη dt ) = ( ϕ ξ ψ ξ ϕ η ψ η ) (f(ϕ,ψ) ) g(ϕ, ψ) 34

35 t (ξ,η) = t (,) Tylor ( ) ϕ(ξ,η) = S ψ(ξ, η) ( ) f(x,y) g(x, y) ( ϕ ϕ ξ ψ η ψ ξ η ( ϕ ξ ψ ξ ϕ η ψ η ( ξη ) + o( ξ + η ) ) = S + o() ) = S + o() ( ) xy = A + o( x + y ) (dξ dt dη dt ( ) ( ) f(ϕ,ψ) ϕψ = A + o( ( ) ξη ϕ g(ϕ, ψ) + ψ ) = AS + o( ξ + η ) ) = (S + o()){as ( ) ξη + o( ( ) ξ + η )} = S ξη AS + o( ξ + η ) A S AS C C C 7. + () (3) (^^;) () (.38) u(α βv) =, v(γu δ) = (u,v) = (,), ( γ δ, α ) αu, δv β u(α βv) = δβ γ (v α β ) + αγ, v(γu δ) = β (u γ δ ) + (.39) () (.4) u(α β u β v) =, v(γ u γ v δ) = (u,v) = (,), (, γ δ ), ( α,), ( β A, B ), A = αγ + β δ, B = αγ β δ, = γ β +γ β, 4. (,) u = αu, v = δv (, δ ) ( ) γ, u = αγ + β δ u, v γ = γ δ u+δ(v+ γ δ αγ +β δ γ ) γ γ αγ δ +β δ γ δ γ, δ ( α,) u β = α(u α ) αβ v, v β β = αγ β δ v 35 β

36 ( ) α αβ β αγ β δ α, αγ β δ β β αγ > β δ αγ < β δ αγ = β δ ( A, B ), u = β A (u A ) β A (v B ( ), v = γ B (u A ) γ B (v B A β ) β A ) B γ γ B, β A + γ B, A B (β γ + β γ ) = {(β A + γ B) 4AB(β γ + β γ )} = {(β A γ B) 4ABβ γ } B = αγ β δ < ( v ) B = αγ β δ >, (β A γ B) 4ABβ γ (β A γ B) 4ABβ γ < αγ > β δ, < αγ > β δ, αγ < β δ B = αγ = β δ v u α 3 β αγ = β δ ( β α,) u = u{( β (u α β ) β v} = α(u α β ) β β α v β (u α β ) β (u α β )v, v = v{γ (u α β ) γ v} = γ (u α β )v γ v α(u β α ) β β v α α(u β α ) + β β v = α v = v{γ (u α β ) γ v} = (γ β β α + γ )v < v > v < 36

37 αγ = β δ ( α,) β (3) (.4) u(α β u β v) =, v(δ γ u γ v) = (,), (, γ δ ), ( α,), ( β A, B ),, A = αγ β δ, B = β δ αγ, = β γ β γ 4 ( = ) (,) u = αu, v = δv (, γ δ ) u = αγ β δ γ u, v = γ δ γ u δ(v δ γ ). αγ β δ, δ αγ > β δ αγ < β δ γ αγ = β δ ( α,) β u = αu, v = β δ αγ v β β δ > αγ β δ < αγ β δ = αγ ( A, B ) u = β A (u A) β A (v B ), v = γ B (u A) γ B (v B A ) ( β A ) β A γ B γ B, T = β A+γ B, (β γ β γ ) AB = AB AB < A > γ > β δ α, B > α δ > γ, > β γ > γ A >, B > A <, B < < β β 3 A,B > < = ( β A γ ) B 4 AB = (β A γ B) + 4β γ AB T <, > T >, > T <, < T >, < = A,B 4 = A,B u,v 37

38 β u + β v = α t AB A,B AB < A = 4 (, γ δ ) v γ u + (v γ δ ) > γ v u + (v γ δ ) = γ u β u = β u β u(v δ γ ) = β { β β u + (v δ γ )} > > γ u > u < () β γ u < u > < B = γ A >,B >, < A <,B <, < A >,B <, <,T <, < A = B = = AB <, = A =,B, = 7. d s dt = g s, d s 4 dt +kds dt g s = x = s, y = ds 4 dt x = y, y = ky + g 4 x ( ) x = y =, g 4 k λ g 4 k λ = λ + kλ g = 4 k ± k + g 4 k ± k + g 4 (, ) t ( ) (s < ) 38

39 k + k + g 4 y (i.e. ) y = x, y (i.e. ) 7. C x = ϕ(t), y = ψ(t), t T C C x ϕ (t), ψ (t) dt C ([], 9 9. ) z C z C C z Cuchy (Cuchy Green ) z C C z z C z C C C P, C C ( ). P U U C, U U+, C C U \ C C U + C ( C = C U +, z U + πi C ζ z dζ = πi C ζ z dζ + πi U + ζ z dζ C z C C U + Cuchy (Cuchy z U + Cuchy θ.) C z C P, C. C U i, i =,,...,N U C t, z C C 39

40 U i \ C Ui U i + U = N i= U i C U \ C = U U +, U = N i= U i, U+ = N i= U+ i, R \ C U Ω, U + Ω + R z Ω, z Ω R \ C 3 Ω C C U C Ω C U z U+ C U i U + i U i U + i C x =, y = b x = x + t, y = y + bt (x,y ) b x, y mod x b > x y = bt = x, b y =, x = b x mod x = y = x mod b x n b mod, n =,,... b n mod t λ = b nλ mod = nλ [nλ], [ ] Guss ( ) Weyl [,] x [,] [,] 7.4 FORTRAN lemniscte.f 7.5 (x(t),y(t)) (x,y ) γ (x,y ) γ (x,y ) (γ Bolzno-Weierstrss. [5], 4.7 ) (x,y ) (x,y ) l l (x,y ), (x,y ) (x,y ) (x,y ) 4

41 C γ U + U U + t γ γ U U = U + γ U U t γ 7.6 U x = y z, y = x + y, z = bx + z(x c) X = x + b, Y = y b, Z = z + b dx = dx dt dt dy = dy dt dt dz = dz dt dt = Y b Z + b = Y Z, = X b + (Y + b) = X + Y, = b(x b) + (Z b)(x b c) = Z(X b c) + bc bc, b + c b, c, ( ) 4