y = y(x) d n y dx n 1.1 dx dy 1.2 y = f( y x ), y + p(x)y = q(x) y(x) =Ce x dx1 p(x 1 ),

Dimensione: px

Iniziare la visualizzazioe della pagina:

Download "y = y(x) d n y dx n 1.1 dx dy 1.2 y = f( y x ), y + p(x)y = q(x) y(x) =Ce x dx1 p(x 1 ),"

Ottavio Sartori
5 anni fa
Visualizzazioni

1 I : I x y = y(x) x y y = dy dx,y = d2 y dx,,y (n) = 2 n F (x, y, y,,y (n) )=0 n d n y dx n : y (n) = f(x, y, y,,y (n ) ) : n n : : y = y(x). y = dy = X(x) Y (y), dx dx dy X(x) :x Y (y) :y y dȳ Y (ȳ) = x d xx( x)+c ( ).2 y = f( y x ), f(t) :t u = y x u y = ux y = u x + u.3 ( ) y + p(x)y =0 p(x) :x y(x) =Ce x dx p(x ), C.4 ( ) y + p(x)y = q(x) p(x),q(x) :x

2 I : 2 y(x) =C(x)e x dx p(x ) C(x) ( ) x x dx y(x) =(C 0 + dx q(x )e 2 p( x 2 ) )e x dx p(x ), C 0.5 P (x, y)dx + Q(x, y)dy =0 P (x, y) + Q(x, y) dy dx =0 P (x, y),q(x, y) :x, y P y = Q x = P (x, y) Φ y dy =0 x, y Φ x dφ = Φ x dx + Φ y Φ Φ(x, y) = x Φ(x, y) =C, x 0 dx P (x,y)+ y C y 0 dy Q(x 0,y ), = Q(x, y) Φ(x, y) x 0,y 0 M(x, y) ( ) (MP) y = (MQ) x MPdx+ MQdy =0.6 (i) y + p(x)y = q(x)y n, (n :, 0) y n u = y n (ii) y + P (x)+q(x)y + R(x)y 2 =0 y y = u + y (iii) ( ) p = y x y = xy + f(y ) dp df (p) (x + dx dp )=0

3 I : 3 dp =0 y = Cx + f(c), dx C : { x + f (p) =0 y = xp + f(p) p

4 I : 4 2 P n (x) =,P n (x),,p (x),p 0 (x) Q(x) n L[y(x)] = P j (x) dj dx j y(x) =y(n) (x)+p n (x)y (n ) (x)+ +P (x)y (x)+p 0 (x)y(x) =Q(x) j=0 y = y(x) n C y (x) + + C n y n (x) = 0 C = = C n = 0 n y (x),,y n (x) y (x) y 2 (x) y n (x) y W (x) = det W, W = (x) y 2 (x) y n (x) y (n ) (x) y (n ) 2 (x) y n (n ) (x) W (x) W (x) 0 {y,,y n } 2. Q(x) =0 Q(x) 0 y(x) =y (x) ( L[y ]=Q ) y(x) =y 0 (x) ( L[y 0 ]=0) y(x) =y (x)+y 0 (x) ( L[y 0 + y ]=Q ) y (x) y 2 (x) C y (x)+c 2 y 2 (x) n ( ) (y,,y n ) n j= C j y j (x) 2.2 P n (x) =,P n (x),,p (x),p 0 (x) ( ) I y = e λx L[e λx ] = f(λ)e λx =0 f(λ) = n P j λ j =0 ( ) λ f(λ) = n j= (λ λ j )=0 j=0

5 I : 5 (a) e λjx, j =,,n e θ = cos θ + i sin θ λ j = λ R + iλ I e λrx cos λ I x e λrx sin λ I x (b) λ = λ j k e λjx, xe λjx, x 2 e λjx,,x k e λjx, λ j = λ R + iλ I k e λrx cos λ I x, xe λrx cos λ I x,, x k cos λ I x e λrx sin λ I x, xe λrx sin λ I x,, x k sin λ I x II * Q(x) =e αx α y = Ce αx C α y = Cxe αx C α m y = Cx m e αx C * Q(x) y y (x),,y n (x) n y(x) =C (x)y (x) + + C n (x)y n (x) W W C (x). C n (x) C n(x) = 0. 0 Q(x) W = det W 0 C j (x) C j(x) = x dx Q(x ) W (x) +C0 j (* x ) ( ) ( ) y (x) y 2 (x) y W = y (x) y 2 (x) W = 2 (x) y 2 (x) W (x) y (x) y (x) C (x) = Q(x)y 2(x) C 2 (x) =Q(x)y 2(x) C 0 C0 2 x y(x) =y (x){ dx Q(x ) y 2(x ) x W (x ) + C0 } + y 2 (x){ dx Q(x ) y (x ) W (x ) + C0 2}

6 I : y = y(x) D = d dx f(d)y(x) =Q(x) f(t) t d2 y 3 dy dx 2 dx +2y =(D2 3D +2)y =(D 2)(D )y Q(x) f(d) f(d) f(d)[ Q(x)] = Q(x) f(d) f(d) Q(x) f(d)y 0(x) =0 y 0 x D Q(x) = dx Q(x ) x xn xn x2 D n Q(x) = dxn dxn dxn 2 dx Q(x ) x Q(x) =eαx dx e αx Q(x ) D α Q(x) =eαx (D α) n D n [e αx Q(x)] f(d) eαx = f(α) eαx (f(α) 0) (D α) n eαx = n! xn e αx f(d) = g(d) (D α), g(d) : degree of g(d) is less than d dα α Q(x) ( k ) Q(x) = D α α D α Q(x) = α k ( D α )j Q(x) j=0

7 I : 7 3 N t x = x (t),x 2 = x 2 (t),,x N = x N (t) d dt x (t) = ẋ (t) =a x (t)+a 2 x 2 (t)+ + a N x N (t)+b (t) d dt x 2(t) = ẋ 2 (t) =a 2 x (t)+a 22 x 2 (t)+ + a 2N x N (t)+b 2 (t) d dt x N(t) = ẋ N (t) =a N x (t)+a N2 x 2 (t)+ + a NN x N (t)+b N (t) b (t),,b N (t) N d dt x(t) = x(t) =A x(t)+ b(t), a a 2 a N x (t) a 2 a 22 a 2N x 2 (t) A =., x(t) =, b(t) = a N a N2 a NN x N (t) b (t) b 2 (t). b N (t) 3. b = 0 x 0 N x(t) =e At x 0, e At = n=0 t n n! An ( ẋ = ax(t) x(t) =e at x 0 ) N N N N x j (t) C N x(t) = C j x j (t) =Φ(t) C, Φ(t) = [ x, x 2,, x N ], C C 2 = j=. Phi(t) Case : A N λ j j =,,N N v j, j =,,N A v j = λ j v j N C N x j (t) =e At v j = e λ jt v j

8 I : 8 Case 2 : A n λ (A λi) v = 0 (A λi) v 2 = v, (A λi) 2 v 2 = 0 (A λi) v 3 = v 2, (A λi) 3 v 3 = 0 (A λi) v n = v n, (A λi) n v n = 0 n x = e At v = e λt e (A λi)t v = e λt v [ x 2 = e At v 2 = e λt e (A λi)t v 2 = e λt I + t ]! (A λi) v 2 = e λt [ v 2 + t v ] [ x 3 = e At v 3 = e λt e (A λi)t v 3 = e λt I + t! x n = e At v n = [ e λt e (A λi)t v n = = e λt (A λi)+t2(a λi)2 2! ] v 3 = e λt [ ] v n + t v n + t2 2! v n tn n! v ] v 3 + t v 2 + t2 2! v 3.2 x = A x + b(t) Φ(t) N x(t) = C j (t) x j (t) =Φ(t) C(t), Φ(t) = [ x, x 2,, x N ], C(t) = j= C (t) C 2 (t). C N (t) x = Φ C +Φ C C =Φ b t = t 0 x = x(t 0 ) t x(t) = G(t, t ) b(t )+ x(t 0 ) t 0 G(t, t ) = Φ(t)Φ (t )=e A(t t ) 3.3

9 I : 9 4 y = y(x) 2 y + q(x)y + r(x)y =0 ( ) (*) q(x) = n=0 q n (x a) n r(x) = n=0 r n (x a) n x = a x = a y = c n (x a) n n=0 c n (n ) k (k+2)(k+)c k+2 + {(m+2)(m+)c m+2 +q k m (m+)c m+ +r k m c m }, k =0,, 2, m=0 c 0 c. Legendre x =0 ( x 2 )y 2xy + n(n +)y =0 2 x = a (*) q(x) = n=0 q n (x a) n r(x) = n=0 r n (x a) n q(x) r(x) (*) x = a y + x a q(x)y + r(x)y =0 (x a) 2 y =(x a) ρ k n=0 c n (x a) n f(ρ + k)c k = (m + ρ)(m + ρ ) + q k m (m + ρ)+r k m }c m, k =0,, 2, m=0 f(ρ) ρ(ρ ) + q 0 ρ + r 0 c m m =0 f(ρ) =0

10 I : 0 c 0 m =, 2, 3, f(ρ + m)c m =(c 0,c,,c m ) f(ρ + m) 0 c m ρ f(ρ) =0 2 ρ ρ 2 () ρ ρ 2 ρ ρ 2 (2) ρ ρ 2 ρ. Bessel x =0 y x y +( ν2 )y =0 x2

Documenti analoghi

6y y = p 2z = q² : q² = 4z 6z = 8c + 6c = a 9 = 8n : 4n = m : m = y + y² = m + 3m = q : 9 = a a = n 7n = z ( 3n) = p + p = q² + q² = b b² = 5 + z =

Verifica n 1 Alunno Data 6y y = p 2z = q² : q² = 4z 6z = 8c + 6c = a 9 = 8n : 4n = m : m = y + y² = m + 3m = q : 9 = a a = n 7n = z ( 3n) = p + p = q² + q² = b b² = 5 + z = m a = n b = 10b³ : 7b = pq pq