3.7 Introduction to the Famous Scientists
237
Problems 3
Prove that the extremal curves of the following basic variational problems can be
included in the extremal curve field (proper or central).
3.1 J [y] =
1
0 (y
2
− 2x y)dx, y(0) = y(1) = 0.
3.2 J [y] =
1
0 (2e
x y + y
2
)dx, y(0) = 1, y(1) = e.
Discuss the extremal properties of the following functionals.
3.3 J [y] =
2
0 (x y
+ y
2
)dx, y(0) = 1, y(2) = 0.
3.4 J [y] =
x 1
0 (y
2
+ 2yy
− 16y
2
)dx, x 1 > 0, y(0) = 0, y(x 1 ) = 0.
3.5 J [y] =
2
−1 y
(1 + x
2 y
)dx, y(−1) = 1, y(2) = 4.
3.6 J [y] =
2
1 y
(1 + x
2 y
)dx, y(1) = 3, y(2) = 5.
3.7 J [y] =
2
−1 y
(1 + x
2 y
)dx, y(−1) = y(2) = 1.
3.8 J [y] =
π
4
0 (4y
2
− y
2
+ 8y)dx, y(0) = −1, y
π
4
= 0.
3.9 J [y] =
2
1 (x
2 y
2
+ 12y
2
)dx, y(1) = 1, y(2) = 0.
3.10 J [y] =
x 1
x 0
1+y
2
y 2 dx, y(x 0 ) = y 0 , y(x 1 ) = y 1 .
3.11 J [y] =
1
0 (y
2
+ y
2
+ 2ye
2x
)dx, y(0) =
1
3
, y(1) =
1
3
e
2 .
3.12 J [y] =
π
4
0 (y
2
− y
2
+ 6y sin 2x)dx, y(0) = 0, y
π
4
= 1.
3.13 J [y] =
x 1
0
dx
y , y(0) = 0, y(x 1 ) = y 1 , x 1 > 0, y 1 > 0.
3.14 J [y] =
x 1
0
dx
y 2 , y(0) = 0, y(x 1 ) = y 1 , x 1 > 0, y 1 > 0.
3.15 Find the second and third variation of the functional J [y] =
1
0 (x y + y
2
− 2y
2 y
)dx.
3.16 Let the functional J [y] =
x 1
x 0
(x
2
+ y
2
+ y
2
)dx, the boundary conditions are
y(x 0 ) = 0, y(x 1 ) = y 1 . Find the solution that the Jacobi equation satisfies the
boundary conditions u(0) = 0, u
(0) = 1.
3.17 Find the extremal curve of the functional J [y] =
1
0 (y
2
− 2yy
+ y
2
)dx,
and indicate on the extremal curve whether the functional can get absolute
maximum (minimum).
3.18 Find the extremal curve of the functional J [y] =
1
0 −2(y
2
− 1)
2 dx, and
discriminate on the extremal curve whether the functional can get absolute
maximum (minimum).
3.19 Find the extremal curve of the functional J [y] =
2
0 (yy
+ y
2
)dx, and discuss
the extremal property, the boundary conditions are y(0) = 0, y(1) = 2.
3.20 Find the extremal curve of the functional J [y] =
(8,2)
(1,1) x
2
3 y
2 dx, and discuss
the extremal property.
3.21 Find the extremal curve of the functional J [y] =
(2,4)
(1,1)
x
3
y 2 dx, and discuss the
extremal property.
3.22 Let u = u(x, y) be the extremal function of the quadratic functional, the
functional
J [u] =
¨
D
( pu
2
x + pu
2
y − 2 f u)dxdy
237
Problems 3
Prove that the extremal curves of the following basic variational problems can be
included in the extremal curve field (proper or central).
3.1 J [y] =
1
0 (y
2
− 2x y)dx, y(0) = y(1) = 0.
3.2 J [y] =
1
0 (2e
x y + y
2
)dx, y(0) = 1, y(1) = e.
Discuss the extremal properties of the following functionals.
3.3 J [y] =
2
0 (x y
+ y
2
)dx, y(0) = 1, y(2) = 0.
3.4 J [y] =
x 1
0 (y
2
+ 2yy
− 16y
2
)dx, x 1 > 0, y(0) = 0, y(x 1 ) = 0.
3.5 J [y] =
2
−1 y
(1 + x
2 y
)dx, y(−1) = 1, y(2) = 4.
3.6 J [y] =
2
1 y
(1 + x
2 y
)dx, y(1) = 3, y(2) = 5.
3.7 J [y] =
2
−1 y
(1 + x
2 y
)dx, y(−1) = y(2) = 1.
3.8 J [y] =
π
4
0 (4y
2
− y
2
+ 8y)dx, y(0) = −1, y
π
4
= 0.
3.9 J [y] =
2
1 (x
2 y
2
+ 12y
2
)dx, y(1) = 1, y(2) = 0.
3.10 J [y] =
x 1
x 0
1+y
2
y 2 dx, y(x 0 ) = y 0 , y(x 1 ) = y 1 .
3.11 J [y] =
1
0 (y
2
+ y
2
+ 2ye
2x
)dx, y(0) =
1
3
, y(1) =
1
3
e
2 .
3.12 J [y] =
π
4
0 (y
2
− y
2
+ 6y sin 2x)dx, y(0) = 0, y
π
4
= 1.
3.13 J [y] =
x 1
0
dx
y , y(0) = 0, y(x 1 ) = y 1 , x 1 > 0, y 1 > 0.
3.14 J [y] =
x 1
0
dx
y 2 , y(0) = 0, y(x 1 ) = y 1 , x 1 > 0, y 1 > 0.
3.15 Find the second and third variation of the functional J [y] =
1
0 (x y + y
2
− 2y
2 y
)dx.
3.16 Let the functional J [y] =
x 1
x 0
(x
2
+ y
2
+ y
2
)dx, the boundary conditions are
y(x 0 ) = 0, y(x 1 ) = y 1 . Find the solution that the Jacobi equation satisfies the
boundary conditions u(0) = 0, u
(0) = 1.
3.17 Find the extremal curve of the functional J [y] =
1
0 (y
2
− 2yy
+ y
2
)dx,
and indicate on the extremal curve whether the functional can get absolute
maximum (minimum).
3.18 Find the extremal curve of the functional J [y] =
1
0 −2(y
2
− 1)
2 dx, and
discriminate on the extremal curve whether the functional can get absolute
maximum (minimum).
3.19 Find the extremal curve of the functional J [y] =
2
0 (yy
+ y
2
)dx, and discuss
the extremal property, the boundary conditions are y(0) = 0, y(1) = 2.
3.20 Find the extremal curve of the functional J [y] =
(8,2)
(1,1) x
2
3 y
2 dx, and discuss
the extremal property.
3.21 Find the extremal curve of the functional J [y] =
(2,4)
(1,1)
x
3
y 2 dx, and discuss the
extremal property.
3.22 Let u = u(x, y) be the extremal function of the quadratic functional, the
functional
J [u] =
¨
D
( pu
2
x + pu
2
y − 2 f u)dxdy