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J[y] =
2
R
0
(y
02
+ 2yy
0
− 16y
2
)dx; y(0) = 0; y(2) = 0.
J[y] =
2
R
0
(xy
0
+ y
02
)dx; y(0) = 1; y(2) = 0.
J[y] =
1
R
−1
1+y
2
y
02
dx; y(−1) = 2; y(1) = 3.
J[y] =
1
R
−1
(y
2
+ y
02
− 2y sin x)dx; y(−1) = 2; y(1) = 3.
J[y] =
1
R
−1
(16y
2
− y
00
2
+ x
2
)dx; y(−1) = 2; y(1) = 3.
J[y] =
1
R
−1
y
02
x
3
dx; y(−1) = 2; y(1) = 3.
J[y] =
1
R
0
(y
2
+ y
02
+ 2ye
x
)dx; y(0) = 1/3; y(1) = e
2
/3.
J[y] =
1
R
−1
(y
2
− y
02
− 2y sin x)dx; y(−1) = 2; y(1) = 3.
J[y, z]=
π/ 2
R
0
(y
02
+ z
02
+2yz)dx;
½
y(0)=1; y(π/2) = 1;
z(0)=0; z(π/2) = −1;
J[y, z]=
R
2
0
(2y
0
z
0
+ y
2
− z
2
+2ze
x
)dx;
½
y(0)=1; y(2) = −1;
z(0)= − 2; z(2) = 1;
J[y, z]=
R
2
0
(2y
0
z
0
− y
2
+ z
2
+2y sin x)dx;
½
y(0)=1; y(2) = −1;
z(0)= − 2; z(2) = 1;
J[y, z]=
R
1
−1
((y + z)
2
− y
02
− z
02
+2xz)dx;
½
y(−1)=2; y(1) = 0;
z(−1)=0; z(1) = 2;
J[y, z]=
R
2
0
((y −z)
2
+ y
02
− z
02
+2z cos x)dx;
½
y(0)=0; y(1) = 1;
z(0)=1; z(1) = 0;
J[y, z]=
R
2
0
(y
2
+4yz+z
2
−y
02
−z
02
+ze
3x
)dx;
½
y(0)=1; y(2) = −1;
z(0)= − 2; z(2) = 1.
J[y, z]=
R
2
−2
(2y
0
z
0
+ y
2
+ z
2
+2ye
−x
)dx;
½
y(−2)=1; y(2) = 3;
z(−2)= − 2; z(2) = 1;
J[y, z]=
R
2
−2
(y
2
+ z
2
− 2y
0
z
0
+2ze
−x
)dx;
½
y(−2)=0; y(2) = 3;
z(−2)=2; z(2) = 1;
R2
12. J[y] = (y 02 + 2yy 0 − 16y 2 )dx; y(0) = 0; y(2) = 0.
0
R2
13. J[y] = (xy 0 + y 02 )dx; y(0) = 1; y(2) = 0.
0
R1 1+y 2
14. J[y] = y 02 dx; y(−1) = 2; y(1) = 3.
−1
R1
15. J[y] = (y 2 + y 02 − 2y sin x)dx; y(−1) = 2; y(1) = 3.
−1
R1 2
16. J[y] = (16y 2 − y 00 + x2 )dx; y(−1) = 2; y(1) = 3.
−1
R1 y 02
18. J[y] = x3 dx; y(−1) = 2; y(1) = 3.
−1
R1
19. J[y] = (y 2 + y 02 + 2yex )dx; y(0) = 1/3; y(1) = e2 /3.
0
R1
20. J[y] = (y 2 − y 02 − 2y sin x)dx; y(−1) = 2; y(1) = 3.
−1
2. Äëÿ ñëåäóþùèõ ôóíêöèîíàëîâ íàéòè ýêñòðåìàëè è ïîñòðîèòü èõ ãðà-
ôèêè:
π/2 ½
R 02 02 y(0)=1; y(π/2) = 1;
1. J[y, z]= (y + z +2yz)dx;
z(0)=0; z(π/2) = −1;
0 ½
R2 0 0 2 2 x y(0)=1; y(2) = −1;
2. J[y, z]= 0 (2y z + y − z +2ze )dx;
z(0)=
½ − 2; z(2) = 1;
R2 0 0 y(0)=1; y(2) = −1;
3. J[y, z]= 0 (2y z − y 2 + z 2 +2y sin x)dx;
z(0)= − 2; z(2) = 1;
½
R1 y(−1)=2; y(1) = 0;
4. J[y, z]= −1 ((y + z)2 − y 02 − z 02 +2xz)dx;
z(−1)=0; z(1) = 2;
½
R2 y(0)=0; y(1) = 1;
5. J[y, z]= 0 ((y − z)2 + y 02 − z 02 +2z cos x)dx;
z(0)=1; z(1) = 0;
½
R2 y(0)=1; y(2) = −1;
6. J[y, z]= 0 (y 2 +4yz+z 2 −y 02 −z 02 +ze3x )dx;
z(0)= − 2; z(2) = 1.
½
R2 y(−2)=1; y(2) = 3;
7. J[y, z]= −2 (2y 0 z 0 + y 2 + z 2 +2ye−x )dx;
z(−2)= − 2; z(2) = 1;
½
R2 y(−2)=0; y(2) = 3;
8. J[y, z]= −2 (y 2 + z 2 − 2y 0 z 0 +2ze−x )dx;
z(−2)=2; z(2) = 1;
9
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