lecture

149 6 6.1 y(x) x = x =1 1+y (x) 2 dx. y(x) 6.1 ( ). xy y (, ) (b, B) y (b, B) y(x) O B y mg (x, y(x)) y(x) 6.1: g m y(x) v =mv 2 /2 mgy(x) v = 2gy(x) 1+y (x) 2 dx 2gy(x) y() =, y(b) =B b x

15 y 6 6.2 ( ). (x, y(x)) mg h l m x O b x y 6.2: y(x) x b G(y) = (m 1+y (x) 2 ) gy(x) dx 1+y (x) 2 dx = l, y() =h, y(b) =h. 6.3 ( ).,. x s(x), y(x) y(x)+ 12 } y (x) 2 dx y(x) 1 y (x) 2 y(x)+ 12 } y (x) 2 dx y(x) s(x), x, 1] 6.1.1 y(x) 1 } 2 dx (6.1)

6.1. 151 F y 1 (x) =x F (y 1 )= y 2 (x) =x 2 F (y 2 )= F F (y 2 ) >F(y 1 ) } 2 1 x 1 dx = 3 x 2 1 } 2 8 dx = 15 F (y) y(x) (6.1), b] f(x) = f(x) dx (6.2) (6.1) F y(x) F y(x) y ȳ(x) =1 F (ȳ) = y(x) y(x) 1 } 2 dx =F (ȳ) ȳ(x) =1 (6.1) 1 O F (ȳ) = F (y 1)= 1 3 F (y 2)= 8 15 1 x 2 y(1) = 2, y(2) = 3 1 y (x) 1 } 2 dx (6.3) y (x) 1, 2] F (6.2) y(x)

152 y 6 F y(x) F 3 2 F (y) > F (ȳ) = y (x) =1 1 y(1) = 2, y(2) = 3 ȳ(x) =x +1 O 1 6.3: ȳ(x) F (ȳ) = y(1) = 2 y(2) = 3 y(x) 2 1 y (x) 1 } 2 dx =F (ȳ) ȳ(x) =x +1 2 x F (y) F (y) y C (6.4) F (y) y C y(x) C 6.4. ȳ(x) C y(x) C F (y) F (ȳ) ȳ(x) (6.4) 6.5. ȳ(x) C ȳ(x) y(x) C F (y) F (ȳ) ȳ(x) (6.4)

6.1. y 153 ȳ(x) y(x) v(x) ε ȳ(x)+εv(x) ȳ(x) ȳ(x) 6.4 y(x), ȳ(x) O 6.4: y(x) = x y(x) = x +.4 cos(4πx) x y ȳ := mx y(x) ȳ(x) x b y(x) ȳ(x) x b y ȳ 1 1 y(x) ȳ(x) 1 1 ( x b) y ȳ y ȳ 1 := mx y(x) ȳ(x) + mx y (x) ȳ (x) x b x b y(x)

154 y 6 y() =, y(1) = 1 (6.5) ȳ(x) (6.5) ȳ(x) v() =, v(1) = v(x) ε ȳ(x)+εv(x) ȳ() + εv() =, ȳ(1) + εv(1) = 1 O 6.5: y(x) = x y(x) = x +.4 sin(4πx) ȳ(x)+εv(x) (6.5) ȳ(x) 6.5 x 3 f(x, y, z) f (x, y(x),y (x)) dx f(x, y, z) y y(x) z y (x) f f(x, y, z) f(x, y, z) 6.6. (1). 6.1 y(x) 1 } 2 dx f(x, y, z) =(y 1) 2 (2). 6.3 y(x)+ 12 } y (x) 2 dx f(x, y, z) =y + 1 2 z2

6.1. 155 f(x, y(x),y (x)) ] f(x, y(x),y (x)) = f y(x) ] y, x y(x) y (x),.. f(x, y(x),y (x)) dx = fy(x)] dx 7.. (1) y(x) =3x 2 (2) y(x) =sin(πx) xy(x)+y (x) 2} dx 6.1.2 1 y(x) v(x) ε y(x) y(x)+εv(x) y(x) y(x) y(x) 2 dx

156 6 F (y + εv) F (y) F (y) y(x) F (y + εv) y(x)+εv(x) y(x) =x 2, v(x) =x 3 y(x)+εv(x) =x 2 + εx 3 F (y + εv) = =2ε ( x 2 + εx 3) 2 ( ) dx x 2 2 dx = 1 3 ε + 1 7 ε2 ( x 4 +2εx 5 + ε 2 x 6 x 4) dx x 5 dx + ε 2 x 6 dx y(x) x 2 x 2 + εx 3 ε ε F (y + εv) F (y) = 1 ε 3 + 1 7 ε

6.1. 157 v ε F (y + εv) F (y) v lim = 1 ε ε 3 6.7. F y(x), v(x) F (y)(v) :=lim ε F (y + εv) F (y) ε y v F F (y + ) F (y) F (y + ) F (y)(v) F (y) O y(x) y(x)+ (x) y 6.6: y(x) v(x) y(x) 2 dx F (y + εv) y(x)+εv(x)} 2 dx = = = y(x)} 2 dx y(x)+εv(x)} 2 y(x)} 2] dx y(x) 2 +2εv(x)y(x)+ε 2 v(x) 2 y(x) 2} dx 2εv(x)y(x)+ε 2 v(x) 2} dx =2ε v(x)y(x) dx + ε 2 v(x) 2 dx (6.6)

158 6 v F (y + εv) F (y) ε =2 ε DF(y)(v) =2 v(x)y(x) dx + ε v(x)y(x) dx v(x) 2 dx y(x) v(x) F 6.8. f(x, y(x),y (x)) dx DF(y)(v) = f y y(x)]v(x)+f z y(x)]v (x)} dx f y 2, f z 3. y(x),v(x) φ(ε) =F (y + εv) 1 F (y + εv) F (y) = ε φ(ε) φ() ε F (y + εv) F (y) φ(ε) φ() DF(y)(v) =lim =lim = φ () ε ε ε ε φ () φ(ε), b], f d dε φ(ε) = d dε fy(x)+εv(x)] dx = d fy(x)+εv(x)] dx (6.7) dε d dε fy(x)+εv(x)] = d dε f(x, y(x)+εv(x),y (x)+εv (x)) (6.8)

6.1. 159. x y(x),y (x),v(x),v (x) (6.8) y(x) =y 1,y (x) =z 1,v(x) =v 1,v (x) =v 2 d f(x, y1 + εv 1,z 1 + εv 2 )) } dε = f y (x, y 1 + εv 1,z 1 + εv 2 )v 1 y 1 + f z (x, y 1 + εv 1,z 1 + εv 2 )v 2 d dε f(x, y(x)+εv(x),y (x)+εv (x)) = f y (x, y(x)+εv(x),y (x)+εv (x))v(x) (6.7) d dε φ(ε) = + f z (x, y(x)+εv(x),y (x)+εv (x))v (x) = f y y(x)+εv(x) ] v(x)+fz y(x)+εv(x)]v (x) ] ] } f y y(x)+εv(x) v(x)+fz y(x)+εv(x) v (x) ε =. F f(x, y) z DF(y)(v) = fy(x)]dx = f y y(x)]v(x)dx = f(x, y(x))dx f y (x, y(x))v(x)dx dx 6.9. (1). y(x) 2 dx f(x, y, z) =y 2 f y =2y f z = DF(y)(v) = 2y(x)v(x) dx

16 6 (2). y(x)+ 12 } y (x) 2 dx f(x, y, z) =y + 1 2 z2 f y =1 f z = z DF(y)(v) = f y y(x)]v(x)+f z y(x)]v (x)} dx = v(x)+y (x)v (x)} dx 8. y(x) v(x) (1). (2). G(y) = xy(x)+y (x) 3} dx 1+y (x) 2 dx 6.1.3 R n f u, v R n f(v) f(u)+ f(u)(v u) u 2 f(u) 6.1. F y(x),v(x) F (y + v) F (y)+df(y)(v) F C y(x) y(x)+v(x) C v(x) F C 6.11. y(x) 2 dx

6.1. 161 F F (y + v) F (y)+df(y)(v) y O y(x) y(x)+v(x) 6.7: (6.6) F (y + v) 2 v(x)y(x) dx + v(x) 2 dx 2 v(x)y(x) dx = DF(y)(v) 6.12. 3 f(x, y, z) x (y, z) g(y, z) =f(x, y, z) x g(y, z) f(x, y, z) 2 3. 6.13. 3 f(x, y, z) f(x, y, z) 2 3 ] f yy (x, y, z) f yz (x, y, z) x, y, z f zy (x, y, z) f zz (x, y, z)

162 6 6.14. F fy(x)]dx = f(x, y(x),y (x))dx. x, b], f(x, y, z) 2 3 F.. f(x, y, z) 2 3 v 1,v 2. f(x, y + v 1,z+ v 2 ) f(x, y, z)+f y (x, y, z)v 1 + f z (x, y, z)v 2 y = y(x), z= y (x), v 1 = v(x), v 2 = v (x) fy(x)+v(x)] fy(x)] + f y y(x)]v(x)+f z y(x)]v (x) fy(x)+v(x)]dx fy(x)] dx + F (y + v) F (y)+df(y)(v) F f y y(x)]v(x)+f z y(x)]v (x)} dx 6.15. (1). x + y(x)2 + y (x) 2 } dx f(x, y, z) =x + y 2 + z 2 F (2). x2 + y(x) 2 + y (x) 2 } dx f(x, y, z) = x 2 + y 2 + z 2 (x, y, z) x 2, 3 F 9. (1). (2). e x y(x)+y (x) 2 } dx x 2 + y(x) 2 + 1+y (x) 2 } dx

6.2. 163 6.2 F (y) := y() =A, y(b) =B f(x, y(x),y (x)) dx (6.9) y y() =A, y(b) =B 6.16. 6.1 1+y (x) 2 dx 2gy(x) y() =, y(b) =B (1). 1. 2. (2). 2 6.17 ( 2 ). DF(y)(v) = f(x, y(x),y (x)) dx f y y(x)] d ] ] b dx f zy(x)]} v(x) dx + f z y(x)]v(x)

164 6. 6.8 DF(y)(v) = = = f y y(x)]v(x)+f z y(x)]v (x)} dx ] b f y y(x)]v(x) dx + f z y(x)]v(x) f y y(x)] d ] dx f zy(x)]} v(x) dx + d dx f zy(x)]} v(x) dx ] b f z y(x)]v(x) 6.2.1 6.18. F (y) := y() =A, y(b) =B f(x, y(x),y (x)) dx (6.1), F ȳ(x) d dx f zy(x)] = f y y(x)] y() =A, y(b) =B ȳ(x) (6.1). ȳ(x) (6.1) F (y) F (ȳ) y() =A, y(b) =B y(x) ȳ() =A, ȳ(b) =B v(x) =y(x) ȳ(x) F (ȳ + v) F (ȳ) v() =v(b) = v(x) ȳ(x) d ( ) dx f zy(x)] = f y y(x)] y() =A, y(b) =B v(x) v() =v(b) =

6.2. 165 2 6.17 DF(ȳ)(v) = = F f y ȳ(x)] d ] ] b dx f zȳ(x)]} v(x) dx + f z ȳ(x)]v(x) F (y + v) F (ȳ)+df(ȳ)(v) =F (ȳ). ȳ (P ). (6.11) 6.19. d dx f zy(x)] = f y y(x)] y() =A, y(b) =B y(x). d dx f zy(x)] = f y y(x)] (6.12) ȳ DF(ȳ)(v) v(x) v DF(ȳ)( ) (6.11) ȳ(x) DF(ȳ)( ) DF(ȳ)(v) = ȳ(x) F (ȳ)(v) = v() =v(b) = v(x) 6.8: y 6.2.2 6.2 ( ). F (y) := y() = 1, y(1) = 2 y(x)+y (x) 2 } dx

166 6 F f(x, y, z) =y + z 2 2 3 6.18 d ( ) dx f zy(x)]} = f y y(x)] y() = 1, y(1) = 2 f y =1, f y y(x)] = 1, f z =2z, ( ) 1 f z y(x)] = 2y (x) d dx 2y (x)} =1 2y (x) =1 2, y (x) = 1 2 x + c 1 y(x) = 1 4 x2 + c 1 x + c 2 c 1,c 2 y() = 1,y(1) = 2 c 2 =1 1 4 + c 1 + c 2 =2 c 1 =3/4,c 2 =1 ȳ(x) = 1 4 x2 + 3 4 x +1 6.2.3

6.2. 167 6.21. F (y) := y() =A, y(b) =B f(x, y(x),y (x)) dx (6.13) ȳ(x) ȳ(x) d ( ) dx f zȳ(x)] = f y ȳ(x)] ȳ() =A, ȳ(b) =B. ȳ(x) F (y) F (ȳ) y() =A, y(b) =B ȳ(x) y(x) v(x) v() =,v(b) = ε ȳ(x)+εv(x) ȳ(x) y O ȳ(x)+εv(x) ȳ(x) x F (ȳ + εv) F (ȳ) ε φ(ε) =F (ȳ + εv) φ φ(ε) φ() ε φ(ε) 1 ε = O ε ȳ(x) ȳ(x)+εv(x) ε φ () =

168 6 DF(ȳ)(v) = (6.14) 2 6.17 v() =v(b) = =DF(y)(v) = f y ȳ(x)] d dx f zȳ(x)]} = f y ȳ(x)] d dx f zȳ(x)]} f y ȳ(x)] d ] dx f zȳ(x)]} v(x) dx = (6.15) ] ] b v(x) dx + f z ȳ(x)]v(x) ] v(x) dx v() =,v(b) = v(x) (6.15) h(x) :=f y ȳ(x)] d dx f zȳ(x)} = x = p h(p) p x h(x) v p (x) 6.9 y f y ȳ(x)] d ] dx f zȳ(x)]} v p (x) dx (6.15) h(x) = f y ȳ(x)] = d dx f zȳ(x)} O h(x) p v p (x) 6.9: v p (x) p x ȳ(x) (6.13) ȳ(x)

6.2. 169 y y v p (x) ȳ(x) ȳ(x)+εv p (x) O p x O p x 6.1: ȳ(x) x = p ȳ(x) x = p 6.1 x = p ȳ(x) 2 v p F (ȳ + εv p ) F (ȳ) lim = DF(y)(v p ) ε ε = f y ȳ(x)] d ] dx f zȳ(x)]} v p (x) dx ȳ F v p x = p ȳ(x) F F (ȳ) ȳ(x) 6.11 6.22. (6.13) F d ȳ(x) dx f zȳ(x)]} = f y ȳ(x)] ȳ() =A, ȳ(b) =B 6.11:

17 6 6.2.4 6.23 (). ( ) F (y) := 1 3 y (x) 3 dx y() = 1, y(1) = 2 d dx f zy(x)]} = f y y(x)] y() = 1, y(1) = 2 ( ) f(x, y, z) = 1 3 z3 f y =, f z = z 2 f y y(x) ] =, fz y(x) ] = y (x) 2 d y (x) 2} = dx y (x) 2 = c 1 c 1 y (x) =± c 1 y (x) =c 1 c 1 y(x) =c 1 x + c 2 c 1,c 2 y() = 1,y(1) = 2 c2 =1 c 1 + c 2 =2 c 1 = c 2 =1 y(x) =x +1

6.2. 171 1. 1. (1). F (y) := y() = 1, y(1) = (2). F (y) := y() =, y(1) = (3). 2xy(x)+y (x) 2} dx 2e x y(x)+y (x) 2 } dx y() = 1, y(1) = (4). F (y) := y() = 1, y(1) = (1 + x 2 )y (x) 2 dx y(x)+ 1+y (x) 2 }dx