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1 IA Last updated: January, n A A Wronskian

2 limit cycle

3 IA hara/lectures/lectures-j.html. y(x) ordinary differential equation, ODE x y = y y = y y + xy y = x y(x) y (x), y (x),..., y n (x) n x (ordinary differential equation, ODE) y = y, y = y. y = y, y = y, y = y. m m- x y(x) y (x), y (x),... t N(t) N (t) = αn(t) α > F = m a x(t) = (x (t), x (t), x (t)) mx = F, mx = F, mx = F F x(t) x(t) t y = F (x, y) F (x, y) F (x, y) = y y = y F (x, y) = y x y = y x y = F (x, y) y = y(x) y = y y(x) = e x y = e x y = e x. C y = Ce x y = x x x / + C

4 IA hara/lectures/lectures-j.html x y = y x = y = y = e x y = Ce x y() = y. C. y (x) = dy = f(y)g(x) (..) dx x y dy f(y) = g(x)dx (..) dy dx ODE dy = g(x) (..) f(y) dx x a b x x b b g(x)dx x y = y(x) a b dy(x) y(b) a f(y(x)) dx dx = dy (..4) y(a) f(y) y(b) y(a) f(y) dy = b a g(x)dx = y(x) y(a) f(y) dy = x a g(u)du. (..5) x u b x x = a y = y(a) y(x) x y = y y() = y y dy = x dx = log y log = (x ) = y(x) = e (x ). (..6) γ t z(t) mz (t) = mg γz (t) t = z = z() = z () =

5 IA hara/lectures/lectures-j.html. y = y(x) dy dx = y = f(x, y), y(x ) = y (..).. ( ) (..) f. f(x, y) (x, y ) D {(x, y) x x < a, y y < b}. f(x, y) D x, y. K D (x, y ) (x, y ) f(x, y ) f(x, y ) K y y Lipschitz (..) y(x ) = y (..) x Remarks. { x x < a b } min a,, M max M. Lipschitz (x,y) D f(x, y) (..) f(x, y) < K (..4) y Lipschitz. Lipschitz. Lipschitz y = y /4 ODE y() = y(x) y = ( 4 x) 4/ 4. ODE y(x) ODE y(x ) = y

6 IA hara/lectures/lectures-j.html 4.. (..) x y(x) = y + f(z, y(z))dz x (..5) (..5) y(x) (..) (..) x x x (..5).. (..5) (..5) y y(z) y x y (x) = y + f(z, y )dz. (..6) x y (x) x x y (..5) y y n y n+ y n+ (x) = y + x y (x) = y + f(z, y (z))dz x (..7) x x f(z, y n (z))dz (n =,,,...) (..8) n y n (x) y(x) lim y n(x) = y(x) (..9) n (..8) n y n y y n+ y (..5) f(x, y) (..9) y(x) (..5) (..) (..9) y n (x) y(x) y n (x) y(x) y n (x).. n (a) x x < a min { } a, b M yn (x) y b

7 IA hara/lectures/lectures-j.html 5 (b) y n+ (x) y n (x) M x x n+ K n /n! n, m > y n+m (x) y n (x) m m y n+i+ (x) y i M x x n+i+ Kn+i (n + i)! M x x i= i= j=n (K x x ) j j! (..) e K x x n n x y n (x) lim n y n (x) (..9) (a), (b) n = y (x) = y (a) (b) x x y (x) y = f(z, y )dz f(z, y ) dz x x. x x M dz = M x x (..) n n + (a) (..) x x x y n+ (x) y = f(z, y n (z))dz f(z, y n (z)) dz M dz = M x x < b (..) x x x x x < a M x x < b (b) y n+ (x) y n+ (x) x { = f(z, yn+ (z)) f(z, y n (z)) } x dz f(z, y n+ (z)) f(z, y n (z)) dz x x x x K y n+ (z) y n (z) dz K M Kn x x n! z x n+ dz = M Kn+ (n + )! z x n+ (..) y n (x) y(x)..4 (..) y () (x), y () (x) (..5) x > x y () (x) y () (x) x { } x f(z, y () (z)) f(z, y () (z)) dz f(z, y () (z)) f(z, y () (z)) dz K Y (x) x x y () (z) y () (z) dz K x x max x z x y () (z) y () (z) (..4) max z: z x < x x y() (z) y () (z), d(x) y () (x) y () (x) (..5) d(x) K x x Y (x) (..6)

8 IA hara/lectures/lectures-j.html 6 d(x) K d(x) K x x d(z)dz K x x d(z)dz K x K x x Y (x)dz = K x x x Y (x), x x K x x Y (x)dz = K! x x Y (x), (..7) n d(x) Kn n! x x n Y (x) (..8) n n d(x) =..5 y + x y + (y ) + y = 4 y, y, y y (x) = y(x), y (x) = y (x), y (x) = y (x) y + x y + (y ) + y = 4, y = y, (..9) y = y ODE n- y = (y, y,..., y n ).. ( ) n y = f (x, y), y = f (x, y),... y n = f n (x, y) (..) f j =,,..., n. f j (x, y) (X, Y ) D {(x, y) x X < a, y j Y j < b}. f j (x, y) D x, y, y,..., y n. K D (x, y) (x, z) f(x, y) f(x, z) K y z K n y j z j Lipschitz (..) j=

9 IA hara/lectures/lectures-j.html 7 y(x ) = Y j =,,..., n y j (x ) = Y j (..) x { x x < a b } min a,, M max M (x, y) D f(x, y) (..) n n n.4 n ODE ODE ODE ODE x n- y (n) (x) + a n y (n ) (x) + a n y (n ) (x) a y (x) + a y(x) = (.4.) y (j) y(x) j- a j Lipschitz y y + y = y() =, y () = y(x) = e λx λ λ λ + = λ =, (.4.) y = e x, e x ODE y (x), y (x) c, c c y (x) + c y (x) ODE

10 IA hara/lectures/lectures-j.html 8 c e x + c e x c, c c + c =, c + c = c =, c = y(x) = e x e x y y y + y = y() =, y () = y () = 4 y(x) = e λx λ λ λ λ + = λ = ±, y = c e x + c e x + c e x y(x) = e x e x + e x n λ λ n + a n λ n + a n λ n a λ + a = (.4.) n λ, λ,..., λ n c e λx + c e λx c n e λnx (.4.4) c c n c, c,..., c n c, c,..., c n y (j) () = d j j =,,..., n c, c,..., c n Λ c, d c + c c n = d, λ c + λ c λ n c n = d, λ c + λ c λ n c n = d,... λ n c + λ n c λ n n c n = d n... c d λ λ... λ n Λ = λ λ... λ n, c = c, d = d... λ n λ n... λ n c n d n n (.4.5) (.4.6) Λ c = d Λ c = Λ d Λ λ i e λx, e λx,..., e λnx y y + y = y() =, y () = y(), y () y(), y () y(), y () c y (x) + c y (x)

11 IA hara/lectures/lectures-j.html 9 λ λ + = λ = y(x) = e x y(x) = x e x c e x + c xe x c = c = ( + x)e x xe x λ, λ λ λ e λx e λx λ λ eλ x e λ x e λx λ λ = x λ eλx n ODE y (n) (x) + a n y (n ) (x) + a n y (n ) (x) a y (x) + a y(x) = (.4.7) λ n + a n λ n + a n λ n a λ + a = (.4.8) λ, λ,..., λ m α, α,..., α m ODE e λ jx, xe λ jx, x e λ jx,..., x α j e λ jx, (j =,,..., m) (.4.9) m α j = n n j= n ODE n y = y y() =, y () = λ = λ = ±i i c e ix + c e ix y(x) = eix e ix i = sin x (.4.) y(x) c, c c, c 4 c cos x + c 4 sin x n n

12 IA hara/lectures/lectures-j.html y (x), y (x),..., y m (x) c y (x) + c y (x) c m y m (x) = x (.4.) c, c,..., c m y (x), y (x),..., y m (x) y (x), y (x),..., y m (x) y (x), y (x),..., y m (x) m n n (.4.9) n (.4.9) n n λ j (.4.6) Λ c = d d = d d Λ Λ c = Λ d Λ Λ etc. Λ.5 n ODE a ij y = a y + a y a n y n, y = a y + a y a n y n,... y n = a n y + a n y a nn y n (.5.) a ij A(x) y j y(x) ODE d y(x) = A y(x) (.5.) dx A y(x) A y(x)

13 IA hara/lectures/lectures-j.html A A.5. A ODE e λx y z y(x) = e λx z ODE(.5.) λ z e λx = A z e λx = A z = λ z (.5.) A λ z z ODE λ z A ODE (.5.) A λ z e λx z λ, λ,..., λ m z, z,..., z m c, c,..., c m c e λ x z + c e λ x z c m e λ mx z m (.5.4) A A A n (.5.4) m = n n y() = y c, c,..., c n c z + c z c n z n = y (.5.5) n n c, c,..., c n n Z c Z [ z, z,..., z n ], c c c c n (.5.6) Z c = y (.5.7) Z c = Z y (.5.8) z, z,..., z n Z c z, z,...

14 IA hara/lectures/lectures-j.html ODE y = ay y(x) = C e ax (.5.) e ax e A = exp(a) e A = exp(a) = n= n! An (.5.9) e x A d dx exa = n= d x n dx n! An = n= x n (n )! An = A n= x n n! An = A e xa (.5.) e xa n- y d ( d dx exa y = dx exa) y = A e xa y (.5.) y(t) = e xa y (.5.) (.5.) (.5.) (.5.) (.5.) y() = y y A y = y + y y () =, y () =. y = y + y [ ] A = [ ] [ ], 4 [ ] [ ] [ ] y (x) = C e x + C e 4x (.5.) y (x) C, C C + C =, C + C = = C =, C = (.5.4) y (x) = e x + e 4x, y (x) = e x + e 4x

15 IA hara/lectures/lectures-j.html [ ] A P P = [ ] B = P AP = B 4 [ ] [ ] [ ] [ ] e xa = e xp BP = P e xb P e x = = e x + e 4x e x + e 4x (.5.5) e 4x e x + e 4x e x + e 4x [ ] [ ] y () (.5.) = y ().5. A A Jordan n n n Au, = λ u,, Au, = λ u, + u,, Au, = λ u, + u,,... Au,p = λ u,p + u,p Au, = λ u,, Au, = λ u, + u,, Au, = λ u, + u,,... Au,q = λ u,q + u,q Au, = λ u,, Au, = λ u, + u,, Au, = λ u, + u,,... Au,q = λ u,q + u,r... Au m, = λ m u m,, Au m, = λ m u m, + u m,, Au m, = λ m u m, + u m,,... Au m,t = λ m u m,t + u m,t p + q + r t = n n λ e λx u,, e λx ( ) u, + tu,, e λ ( x u, + tu, + t u ),, e λ ( x u,4 + tu, + t u, + t! u,),..., e λ x ( u,p + tu,p + t u,p + t! u,p tp (p )! u ), (.5.6) p λ p q q λ p r r λ m n A.6 n ODE y = a y + a y + a y a n y n, y = a y + a y + a y a n y n,... y n = a n y + a n y + a n y a nn y n (.6.)

16 IA hara/lectures/lectures-j.html 4 a ij x a ij (x) A(x) d y(x) = A(x) y(x) (.6.) dx A x.6. ODE d y(x) = A(x) y(x) (.6.) dx y(x), z(x) c, c c y(x) + c z(x) (.6.) (.6.) m y (j) (x) j =,,..., m c, c,..., c m m j= c j y (j) (x) (.6.) (Principle of Superposition) ODE.6. m n ODE n y () (x), y () (x),..., y (m) (x) c y () (x) + c y () (x) c m y (m) (x) x (.6.4) c = c =... = c m = n ODE n y () (x), y () (x),..., y (n) (x) e λx.6. Wronskian n n y () (x), y () (x),..., y (n) (x) n Wronski Ŵ (x) ij y (j) i Ŵ (x) [ ] y () (x), y () (x),..., y (n) (x), (.6.5)

17 IA hara/lectures/lectures-j.html 5 Ŵ Wronskian W W (x) = det Ŵ (x). y () (x), y () (x),..., y (n) (x) x W (x) = det Ŵ (x) W (x) (.6.4) c, c,..., c n (.6.4) x x W (x).6. x = x W (x ) x W (x) W (x ) = x W (x) = d dx W (x) = d [ dx det y (), y (),..., y (n)] = tra(x) W (x) (.6.6) tr W (x) ( x ) W (x) = W (x ) exp tra(t) dt x (.6.7) W (x ) W (x).7 n ODE y = a y + a y + a y a n y n + b, y = a y + a y + a y a n y n + b,... y n = a n y + a n y + a n y a nn y n + b n (.7.) a ij b j x a ij A(x) b j b(x) d dx y(x) = A(x) y(x) + b(x) (.7.) b b.7..

18 IA hara/lectures/lectures-j.html 6 (.7.) y () (x) y () (x) z(x) y () (x) y () (x) z(x) d z(x) = A(x) z(x). (.7.) dx z z + y y z y z + y y.7..8,. y + y = x y() = x y = a + a x a + a + a x = x a =, a = y(x) = x y + y = y(x) = Ce x y(x) = f(x) e x f f (x) e x e x f(x) + f(x) e x = x = f (x) = x e x (.7.4) f f(x) = g(x)e x dx = (x ) e x + const (.7.5) y(x) = x y(x) = Ce x + x C C = y(x) = e x + x C f(x) x.8.6. y y + y = x, y() =, y () = y(x) = a + a x x

19 IA hara/lectures/lectures-j.html 7 y y + y = a x + a a = x a = a = a = 4 y y + y = y(x) = C e x + C e x y(x) = C (x)e x C C = xe x C = z z z = xe x z = C (x)e x C = xe x C = ( x + 4 )e x C = z = ( x + 4 )e x C = ( x + 4 )e x y(x) = x + 4 y(x) = C e x + C e x + x + 4 C, C C =, C = 5 4 y(x) = ex ex + x + 4 n n C z = C.7..6 n ODE d dx y(x) = A(x) y(x) + b(x) (.7.6) n z () (x), z () (x),..., z (n) y(x) = c (x) z () (x) + c (x) z () (x) c n (x) z (n) (x) = Z(x) c(x) (.7.7) Z(x) [ z (), z (),..., z (n)], c(x) = c c c n (.7.8) c (x), c (x),..., c n (x) d dx y = c z () + c z () c n z (n) d + c dx z () d + c dx z () d c n dx z (n) (.7.9) d dx z (j) = A z (j) (j =,,..., n) (.7.)

20 IA hara/lectures/lectures-j.html 8 c z () + c z () c n z (n) = b (.7.) c, c,..., c n (.7.) c, c,..., c n (.7.) c [ z (), z (),..., z (n)] c = b (.7.) c n d dx c = c c = Z b Z Z [ z (), z (),..., z (n)] (.7.) c n c c, c,..., c n (.7.) c c(x) = x y(x) = Z(x) c(x) = Z(x) Z (u) b(u)du (.7.4) x Z (u) b(u)du (.7.5) n Z.6. m k F cos(ωx) my + cy + ky = F cos(ωx) (.7.6) m, c, k, F F = ODE λ + cλ + k = λ = c m ± c 4mk m (.7.7) α, β y = c e αx + c e βx α = β (c + c x)e αx m, c, k.6 c = c > x

21 IA hara/lectures/lectures-j.html 9 c > 4mk c < 4mk c = 4mk F >. c = ω = k/m ω = ω ω ω ω ω ω ω c > ω

22 IA hara/lectures/lectures-j.html. t f(t) F (s) f F (s) f(t) e st dt. (..) s f s f F F f F f F L(f) L(f)(s) = F (s) = f(t) e st dt. (..) F L(f) F F f F f = L (F ) f(t) = e at Re s > Re a L(f)(s) = f(t) = sin ωt Re s L(f)(s) = e at e st e (s a)t dt = lim = t s a s a. (..) sin(ωt) e st ω {ω cos(ωt) + s sin(ωt)}e st ω dt = lim t ω + s = ω + s. (..4) Re s < ω ω sin(ωt) + s.. ( ) f, g a, b L(af + bg) = al(f) + bl(g) (..5) cosh x 5 6 cos x, sin x

23 IA hara/lectures/lectures-j.html.. ( ) f(t) F (s) e at f(t) F (s a) L{e at f(t)} = F (s a) = L{f(t)}(s a), e at f(t) = L {F (s a)} (..6) s p.8, p.7 p.5 55 s F (s) = f(t)e st dt s = s Re s > Re s s f(t)e st dt Re s > σ Re s < σ (..7) σ σ = ± σ Re s > σ F (s) s d n ds n F (s) = ( t) n f(t) e st dt (..8) F (s) f(t).. ( ) F (s) Re s > σ F (s) f = L (F ) s > σ f(t + ) + f(t ) = πi s+i s i F (s)e st ds (..9) e st t e st s

24 IA hara/lectures/lectures-j.html L(e at ) = s a s > a L ( s a ) = eat σ = a s > a s +i s i s a est ds = s + is a e(s +is )t i ds = e s t t s + i(a s ) eis ds (..) = e s t π i e (a s )t = π i e at (..) πi e at.. p.5 55 L (F ) pp.5 55 f F (s) L (f).. f (t) f (t) F (s) f = f f (t) e st dt = F (s) = f (t) e st dt = {f (t) f (t)} e st dt = (..) σ Re s > σ g(t)e st dt = g(t) = (..) g(t) = (..) s g(t) t n e st dt = (..4) n a n ( = a n g(t) t n e st dt = a n t n) g(t) e st dt = (..5) n= a n n= a n t n = h(t) (..5) h(t) g(t) h(t) e st dt = h(t)e st h(t) n= g(t) h(t) dt = (..6) h h g = t g(t) = p.

25 IA hara/lectures/lectures-j.html. f(t). L(f )(s) = sl(f)(s) f(), L(f )(s) = s L(f) sf() f () (..) L(f (n) )(s) = s n L(f) s n f() s n f () f (n ) (). (..) { t L } f(τ)dτ = s F (s), L { } s F (s) = p.,, 6 t f(τ)dτ (..) y y + y = y() = a, y () = b s Y (s) sy() y () {sy (s) y()} + Y (s) = s (..4) (s s + )Y (s) = s + (s )a + b = + as + (b a) (..5) s Y (s) = as + (b a) (s ) + s (s ) (..6) y(t) = + (a )e t + (b a + )te t (..7) as + (b a) + s(s ) (s ) = s + s + (s ) + a s + b a (s ) (..8) p.7 y(t) = + ( )e t + te t + ae t + (b a)t e t = + (a )e t + (b a + )te t (..9) (s a) n s n e at y(t) y y + y = x cos x y() = a, y () = b s Y (s) sy() y () {sy (s) y()} + Y (s) = s (s + ) (..) (s s + )Y (s) = s (s + ) + (s )a + b = s (s + (s )a + b (..) + )

26 IA hara/lectures/lectures-j.html 4 Y (s) = s (s )a + b (s + ) + (s )(s ) (s )(s ) = (s + ) (s )a + b (s + ) + (s ) (s )(s ) Y (s) = 5 s s 5 s s + s 5 (s + ) 5 (s + ) + a b s + b a s (..) (..) p.7 p.5 55 (s a) s s + cos t, sin t, t sin t, t cos t y(t) = 5 et 5 cos t 6 5 sin t 5 t sin t 5 (sin t t cos t) + (a b)et + (b a)e t ( = (a b)e t + b a + ) e t 7 cos t sin t + t cos t t sin t (..4) a = b =. u(x) (t < ) u(t) = (t > ) (..) t = a u(t a) (t < a) u(t a) = (..) (t > a) u(t a) t a u a (t) = u(t a) u a t a f f(t)u(t a) f(t) t < a f(t a)u(t a) f(t) t > a u s L{u(t a)} = s e as (a > ) (a ) (..)

27 IA hara/lectures/lectures-j.html 5 a > L{f(t a) u(t a)} = e as F (s) L {e as F (s)} = f(t a) u(t a). (..4) δ(t) f f(t) δ(t) dt = f() (..5) u a δ a (t) = δ(t a) f(t a) δ(t) dt = f(a). (..6) (..5) (..5) f() δ(t) t t = t = δ(t) t = ϕ n (t) n f n n n/ ( t < /n) ϕ n (t) = π e nt, ϕ n (t) = (..7) ( ) δ(t a) = d u(t a). (..8) dt f(t) f(t) δ(t a) = = a f(t) d ] [f(t) dt u(t a) = u(t a) f (t) u(t a) dt [ ] f (t) dt = f(t) = f( ) + f(a) = f(a) (..9) a e as (a > ) L{δ(t a)} = (a < ) (..)

28 IA hara/lectures/lectures-j.html 6.. f(t) t f(t) f(t) t T f(t) T (f) T T c, c f, f g(t) f(t) T (c f + c f ) = c T (f ) + c T (f ) (..) f(t)g(t)dt (..) g δ(t a) (..6) f f(a) (..) (..7) ϕ n δ(t a)f(t) dt = f(a) = lim n ϕ n (t a)f(t)dt (..4) δ(t a) = lim ϕ n (t a) f(t) (..) δ(t a) = u (t a) (..8) (..9) f(t) f (a) f (a) f(a) (..5)

29 IA hara/lectures/lectures-j.html 7 δ (t a) δ (t a) δ (t a)f(t) = [ ] δ(t a)f(t) δ(t a) f (t) = f (a) (..6) (..5) f(t) ( ) n f (n) (a) f (n) (a) f(a) n (..7) n δ (n) (t a).4 f. F f F L{tf(t)} = F (s), L {F (s)} = tf(t) (.4.) { f(t) } L = t s F (s )ds L { p. s F (s )ds } = f(t) t (.4.) p. p..5 f, g h(t) f, g convolution f g (f g)(t) = t f(t τ)g(τ)dτ = t f(τ)g(t τ)dτ (.5.) (, t) (, ) t (, t) f, g τ t ρ(y) x y x 4π x y ρ(y) φ(x) = ρ(y) dy (.5.) 4π R x y

30 IA hara/lectures/lectures-j.html 8 ρ x y f, g, h F, G, H H(s) = F (s)g(s) (.5.)

31 IA hara/lectures/lectures-j.html 9 I n t y (t), y (t),..., y n (t) y (t) = f (y (t), y (t),..., y n (t)), y (t) = f (y (t), y (t),..., y n (t)),..., y n(t) = f n (y (t), y (t),..., y n (t)) (..) t t y (t),..., y n (t) y(t) (y (t), y (t),..., y n (t)) f i f( y(t)) = (f, f,..., f n ) d dt y(t) = f( y(t)) (..). I. (..) y, y,..., y n n- t y(t) = (y (t), y (t),..., y n (t)) y(t) n- t y(t) = (y (t), y (t),..., y n (t)) n- y (t) (y (t), y (t),..., y n(t)) t y(t) f( y(t)) t f( y(t)) f( y(t)) y(t) t y(t) f( y(t))

32 IA hara/lectures/lectures-j.html x = x y = x x = y (..) y = x x = y y = x γy (..) γ =.,.,. γ = γ < x() =., y() =

33 IA hara/lectures/lectures-j.html γ =., γ =, γ =. γ Duffing x = y (..) y = x αx α = α > α < α =

34 IA hara/lectures/lectures-j.html (, ) (±, ) van der Pol p.89 x = y y = x + µ( x )y (..4) µ =.

35 IA hara/lectures/lectures-j.html µ µ =

36 IA hara/lectures/lectures-j.html 4 Lotka-Volterra µ =. x = x( y) y = µy(x ) (..5) µ =.4

37 IA hara/lectures/lectures-j.html 5. limit cycle I. Lotka-Volterra (, ) fixed point van del Pol limit cycle limit cycle Limit cycle f( y) = I.

38 IA hara/lectures/lectures-j.html 6. van del Pol µ =. µ =. t t t.. P ODE P t P P ϵ > δ > y() P < δ y() y(t) P < ϵ P P t t t van der Pol Lotka-Volterra (, )

39 IA hara/lectures/lectures-j.html 7 Duffing α < (, ) (± /α, ) (± /α, ) Duffing y =, x + αx = (..) α > (, ) α < (, ) (± /α, ) α < α < α = α < (, ) (±, ) (, ) (, ) x, y ODE x, y Duffing x = y (..) y = x ODE (x, y) (, ) x(t) = + u(t), y(t) = + v(t) = v(t) (..) u, v u, v ODE u, v u = x = y = v (..4) v = y = x + x = u + u + y u, v u = v v = u (..5) [ ] [ ] c e t + c e t (..6) x αx

40 IA hara/lectures/lectures-j.html 8 e t e t c (u, v) (x, y) (, ) c ±(, ) (, ) ± (, ± ) Duffing

December 28, 2018

December 28, 2018 e-mail : kigami@i.kyoto-u.ac.jp December 28, 28 Contents 2............................. 3.2......................... 7.3..................... 9.4................ 4.5............. 2.6.... 22 2 36 2..........................